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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>



<b>thus </b><a NAME="E1:57"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V12">modular</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V12">modular</a> implies <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is <a href="lattices.html#V12">modular</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</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#K3">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D12">LATTICES:def 12</a><br/>

<b>assume </b><a NAME="E2:57_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</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#K3">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D12">LATTICES:def 12</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="filter_1.html#K7" target="_self">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D12">LATTICES:def 12</a><br/>

<b>assume </b><a NAME="E3:57_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<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>, <font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:57_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:57_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</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>2</sub></font><b> such that </b><br/><a NAME="E9:57_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E10:57_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E3:57_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="filter_1.html#E5:57_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="filter_1.html#E9:57_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self">Th37</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E7:57_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="filter_1.html#E9:57_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E5:57_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E1:57_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="filter_1.html#E10:57_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E2:57_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="filter_1.html#E10:57_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="filter_1.html#E9:57_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="filter_1.html#E5:57_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="filter_1.html#E7:57_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self">Th36</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:57"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="filter_1.html#K7" target="_self">:]</a></span> st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</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#K3">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D12">LATTICES:def 12</a><br/>

<b>thus </b><a NAME="E3:57"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V12">modular</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D12">LATTICES:def 12</a><br/><b>let </b><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 <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E3:57_2"/>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:57_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E4:57_2"/><b>then </b>
( <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p2"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p2"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E2:57"><i><font color="Green">E36</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self">Th36</a></i>;<br/>
<b>hence </b><a NAME="E5:57_2"/>
<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#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</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> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D12">LATTICES:def 12</a><br/><b>let </b><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 <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E4:57"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E6:57"/>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E4:57"><i><font color="Green">E37</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E7:57"/><b>then </b>
( <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p2"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p2"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E2:57"><i><font color="Green">E36</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self">Th36</a></i>;<br/>
<b>hence </b><a NAME="E8:57"/>
<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#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</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> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div>
