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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span>;<br/>









<a NAME="E1:69"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="lattices.html#NM1">Lattice</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>





<b>thus </b><a NAME="E1:69_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:69_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<a NAME="E4:69_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="filter_2.html#K5" target="_self">.:</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="filter_2.html#K5" target="_self">.:</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E5:69_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span>
 <b>by </b><i><a class="txt" href="filter_2.html#E3:69_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:69_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span><b> such that </b><br/><a NAME="E3:69_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<a NAME="E4:69_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E5:69_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="filter_2.html#E3:69_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:69"/><i><font color="Green">E29</font></i>: 
(  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="lattice2.html#K1">.:</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K7" target="_self">.:</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K7" target="_self">.:</a>  &amp;  <a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a>  <a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a>  <a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
;<br/>
<a NAME="E3:69"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K5">"/\"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E59"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:69"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_2.html#K20" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K5">"/\"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="filter_2.html#K20" target="_self">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="filter_0.html#K5">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="filter_2.html#K20" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="filter_0.html#K5">"/\"</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="filter_2.html#E1:69"><i><font color="Green">E28</font></i></a>, <a class="txt" href="filter_2.html#E2:69"><i><font color="Green">E29</font></i></a></i>;<br/>


</div>
