<?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    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:52"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="filter_0.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:52"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="filter_0.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:52"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K4">"/\"</a> <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#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="filter_0.html#E2:52"><i><font color="Green">E36</font></i></a>, <a class="txt" href="filter_0.html#E4:52"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E7:52"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:52_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:52_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:52_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  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/>
<a NAME="E9:52"/><i><font color="Green">E38</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 <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:52_2"/><b>then </b><i><font color="Green">E39</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
;<br/>
<b>assume </b><a NAME="E3:52_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E4:52_2"/><b>then </b><i><font color="Green">E40</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:52_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="filter_0.html#E2:52_2"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:52_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="filter_0.html#E4:52_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D7">LATTICES:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D7">LATTICES:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D7">LATTICES:def 7</a></i>
;<br/>
<a NAME="E10:52_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_0.html#E6:52_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="filter_0.html#E8:52_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="lattices.html#T27">LATTICES:27</a></i>;<br/>
<a NAME="E11:52_2"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_0.html#E6:52_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="filter_0.html#E8:52_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="filter_0.html#D1" target="_self">Def1</a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E12:52_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="filter_0.html#E9:52_2"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:52"/>
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 <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:52_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:52_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:52_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>assume </b><a NAME="E5:52_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E6:52_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E4:52_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E7:52_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="filter_0.html#E4:52_3"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="filter_0.html#E9:52"><i><font color="Green">E38</font></i></a>, <a class="ref" href="filter_0.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E12:52"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:52_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:52_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<b>hence </b><a NAME="E4:52_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="filter_0.html#E4:52"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:52"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:52_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E3:52_5"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<b>hence </b><a NAME="E4:52_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="filter_0.html#E2:52"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:52"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E12:52"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E15:52"/>
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="filter_0.html#D5" target="_self">Def5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>10</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="E16:52"/>
<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#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E18:52"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E20:52"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="filter_0.html#E18:52"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E21:52"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E22:52"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E23:52"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 
<b>by </b><i><a class="txt" href="filter_0.html#E20:52"><i><font color="Green">E41</font></i></a>, <a class="ref" href="filter_0.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E24:52"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 <b>by </b><i><a class="txt" href="filter_0.html#E18:52"><i><font color="Green">E40</font></i></a>, <a class="txt" href="filter_0.html#E20:52"><i><font color="Green">E41</font></i></a>, <a class="ref" href="filter_0.html#T9" target="_self">Th9</a></i>;<br/>


</div>
