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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:12"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:12_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>
<a NAME="E3:12_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lopclset.html#T6" target="_self">Th6</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>7</sub></font> <a href="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lopclset.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="lopclset.html#K3" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lopclset.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lopclset.html#K2" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lopclset.html#K3" target="_self">/\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="lopclset.html#K2" target="_self">\/</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="lopclset.html#E3:12_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lopclset.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lopclset.html#E2:12_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lopclset.html#T5" target="_self">Th5</a></i>
;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="lopclset.html#T7" target="_self">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="lopclset.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:12"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="lopclset.html#T5" target="_self">Th5</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>

<b>hence </b><a NAME="E3:12_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> = 
<a href="xboole_0.html#K1">{}</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="lopclset.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>

;<br/>

<b>hence </b><a NAME="E3:12_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:12"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V13">lower-bounded</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green">E19</font></i>: 
 <a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V14">upper-bounded</a>
 
<b>by </b><i><a class="txt" href="lopclset.html#E4:12"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lattices.html#D14">LATTICES:def 14</a></i>;<br/>
<a NAME="E8:12"/><i><font color="Green">E20</font></i>: 
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E5:12"><i><font color="Green">E17</font></i></a>, <a class="txt" href="lopclset.html#E6:12"><i><font color="Green">E18</font></i></a>, <a class="ref" href="lattices.html#D16">LATTICES:def 16</a></i>;<br/>
<a NAME="E9:12"/><i><font color="Green">E21</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E4:12"><i><font color="Green">E16</font></i></a>, <a class="txt" href="lopclset.html#E7:12"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lattices.html#D17">LATTICES:def 17</a></i>;<br/>
<b>thus </b><a NAME="E10:12"/>
 <a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V15">bounded</a>
 <b>by </b><i><a class="txt" href="lopclset.html#E6:12"><i><font color="Green">E18</font></i></a>, <a class="txt" href="lopclset.html#E7:12"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D20">LATTICES:def 20</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="lattices.html#NM6">01_Lattice</a> <b>by </b><i><a class="txt" href="lopclset.html#E6:12"><i><font color="Green">E18</font></i></a>, <a class="txt" href="lopclset.html#E7:12"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>
<a NAME="E12:12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="lopclset.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E3:12_4"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>5</sub></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">c<sub>8</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#T5" target="_self">Th5</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="lopclset.html#E9:12"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:12_4"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<i><font color="Green">E24</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lopclset.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="lopclset.html#E8:12"><i><font color="Green">E20</font></i></a>, <a class="txt" href="lopclset.html#E4:12_4"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E6:12_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="lopclset.html#E3:12_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="lopclset.html#E5:12_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:12"/>
(  <a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V16">complemented</a> &amp;  <a href="lopclset.html#K6" target="_self">OpenClosedSetLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V11">distributive</a> )
 <b>by </b><i><a class="txt" href="lopclset.html#E1:12"><i><font color="Green">E15</font></i></a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>


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