<?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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:109"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="filter_0.html#V3">implicative</a>
 ;<br/>

<a NAME="E2:109"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="filter_0.html#NM2">I_Lattice</a>
 
;<br/>
<b>assume </b><a NAME="E3:109"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span>;<br/>
<a NAME="E4:109"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="filter_2.html#T73" target="_self">Th73</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7">FILTER_0:def 7</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self">Th69</a></i>;<br/>
<a NAME="E6:109"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E3:109"><i><font color="Green">E47</font></i></a>, <a class="txt" href="filter_2.html#E4:109"><i><font color="Green">E48</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self">Th63</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E7:109"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E8:109"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E3:109"><i><font color="Green">E47</font></i></a>, <a class="ref" href="filter_0.html#T7">FILTER_0:7</a>, <a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span> <b>by </b><i><a class="txt" href="filter_2.html#E3:109"><i><font color="Green">E47</font></i></a>, <a class="txt" href="filter_2.html#E4:109"><i><font color="Green">E48</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self">Th63</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<a NAME="E10:109"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E2:109"><i><font color="Green">E46</font></i></a>, <a class="txt" href="filter_2.html#E6:109"><i><font color="Green">E49</font></i></a>, <a class="ref" href="filter_2.html#T74" target="_self">Th74</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D12">LATTICES:def 12</a></i>;<br/>
<a NAME="E11:109"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="filter_2.html#E1:109"><i><font color="Green">E45</font></i></a>, <a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a></i>;<br/>
<a NAME="E12:109"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="filter_2.html#E6:109"><i><font color="Green">E49</font></i></a>, <a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/>
<a NAME="E13:109"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T2">FILTER_0:2</a></i>;<br/>
<b>hence </b><a NAME="E14:109"/>
<font color="Maroon">c<sub>6</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>7</sub></font>
 <b>by </b><i><a class="txt" href="filter_2.html#E10:109"><i><font color="Green">E50</font></i></a>, <a class="ref" href="filter_2.html#T75" target="_self">Th75</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self">Th69</a></i>;<br/>
<b>assume </b><a NAME="E16:109"/>
<font color="Maroon">c<sub>6</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>7</sub></font>
 ;<br/>

<a NAME="E17:109"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="filter_2.html#T75" target="_self">Th75</a></i>;<br/>
<a NAME="E18:109"/><b>then </b>
( <font color="Maroon">c<sub>8</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>9</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E3:109"><i><font color="Green">E47</font></i></a>, <a class="txt" href="filter_2.html#E4:109"><i><font color="Green">E48</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self">Th63</a>, <a class="ref" href="filter_2.html#T74" target="_self">Th74</a></i>;<br/>
<a NAME="E19:109"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E1:109"><i><font color="Green">E45</font></i></a>, <a class="txt" href="filter_2.html#E3:109"><i><font color="Green">E47</font></i></a>, <a class="txt" href="filter_2.html#E4:109"><i><font color="Green">E48</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self">Th63</a>, <a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<a NAME="E20:109"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T1">FILTER_0:1</a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>
<a NAME="E21:109"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T27">LATTICES:27</a></i>;<br/>
<b>hence </b><a NAME="E22:109"/>
<font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="filter_2.html#T75" target="_self">Th75</a></i>;<br/>


</div>
