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

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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>
<a NAME="E1:34"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)<br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E2:34"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)<br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:34"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="lattices.html#V10">Lattice-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D4">LATTICES:def 4</a>,<a class="ref" href="lattices.html#D10">LATTICES:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_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> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</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#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:34_1"/>
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 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K1">"\/"</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#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D5">LATTICES:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><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 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</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>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:34_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:34_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p0">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="tdlat_1.html#E2:34_1_2"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p0">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="tdlat_1.html#E3:34_1_2"><i><font color="Green">E34</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:34_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D8">LATTICES:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3"><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> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:34_1_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E3:34_1_3"/>
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:34_1_3"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:34_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E6:34_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E4:34_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:34_1_3"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E8:34_1_3"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/>
<a NAME="E9:34_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E10:34_1_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> )
 
;<br/>
<a NAME="E11:34_1_3"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#D6">TOPS_1:def 6</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p0">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="tdlat_1.html#E2:34_1_3"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p0">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E4:34_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p0">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E8:34_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E11:34_1_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:34_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D6">LATTICES:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_4"><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> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</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#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:34_1"/>
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 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K2">"/\"</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#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D7">LATTICES:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_5"><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 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</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>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:34_1_5"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E3:34_1_5"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p0">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="tdlat_1.html#E2:34_1_5"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p0">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="tdlat_1.html#E3:34_1_5"><i><font color="Green">E34</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:34_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D9">LATTICES:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_6"><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> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:34_1_6"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:34_1_6"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:34_1_6"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:34_1_6"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E6:34_1_6"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E4:34_1_6"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:34_1_6"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E8:34_1_6"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T45">TOPS_1:45</a></i>;<br/>
<a NAME="E9:34_1_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E10:34_1_6"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> )
 
;<br/>
<a NAME="E11:34_1_6"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#D6">TOPS_1:def 6</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p0">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="tdlat_1.html#E2:34_1_6"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p0">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E4:34_1_6"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p0">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="tops_1.html#T46">TOPS_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E8:34_1_6"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E11:34_1_6"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) as   <a href="lattices.html#NM1">Lattice</a> ;<br/>
<a NAME="E5:34"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="lattices.html#V15">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="lattices.html#V13">lower-bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15">LATTICES:def 15</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_2_1"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E2:34_2_1"/><b>then </b>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</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 <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E5:34_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>

;<br/><br/>

<b>hence </b><a NAME="E7:34_2_1"/>
<font color="Maroon">c<sub>5</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>4</sub></font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:34_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="lattices.html#V14">upper-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_2_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E2:34_2_2"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<b>then 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 <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E5:34_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<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="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="tdlat_1.html#E5:34_2_2"><i><font color="Green">E33</font></i></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="E7:34_2_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/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="lattices.html#NM6">01_Lattice</a> ;<br/>
<a NAME="E7:34"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="lattices.html#V16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_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 <font color="Maroon">c<sub>4</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a><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="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:34_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:34_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> )
 
;<br/>
<a NAME="E4:34_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E5:34_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E7:34_3"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E8:34_3"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</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 <font color="Maroon">c<sub>4</sub></font> ;<br/>
<a NAME="E10:34_3"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> holds the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:34_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:34_3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> )
 
;<br/>
<a NAME="E4:34_3_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#D6">TOPS_1:def 6</a></i>;<br/>
<b>thus </b>the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E4:34_3_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E10:34_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="lattice2.html#T25">LATTICE2:25</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a><br/>

<b>hence </b><a NAME="E12:34_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E13:34_3"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E14:34_3"/><b>then </b>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</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 <font color="Maroon">c<sub>4</sub></font> ;<br/>
<a NAME="E16:34_3"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> holds the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:34_3_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:34_3_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> )
 
;<br/>
<a NAME="E4:34_3_3"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#D6">TOPS_1:def 6</a></i>;<br/>
<b>thus </b>the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>10</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E1:34"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T8">TOPS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T8">TOPS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E4:34_3_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E17:34_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E2:34"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_1.html#E17:34_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E16:34_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lattice2.html#T22">LATTICE2:22</a></i>
;<br/>

<b>hence </b><a NAME="E19:34_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:34"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1" target="_self">Domains_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>


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