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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM2">D_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="E1:33"/><i><font color="Green">E28</font></i>: 
not <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">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E2:33"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="openlatt.html#E1:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="openlatt.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E3:33"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="ordinal1.html#V6">c=-linear</a> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="openlatt.html#T20" target="_self">Th20</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:33"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> not <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="openlatt.html#E2:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lattice4.html#T3">LATTICE4:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="openlatt.html#E5:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="openlatt.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E7:33"/><i><font color="Green">E31</font></i>: 
( not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="openlatt.html#E5:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="openlatt.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E8:33"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="openlatt.html#E5:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="openlatt.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E9:33"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="filter_0.html#V2">prime</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="filter_0.html#D6">FILTER_0:def 6</a><br/>

<b>thus </b><a NAME="E1:33_1"/>
(  not <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span>;<br/>
<a NAME="E2:33_1_1"/><i><font color="Green">E34</font></i>: 
(  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></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="E3:33_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="lattice4.html#T6">LATTICE4:6</a></i>;<br/>
<b>consider </b><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="E5:33_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="openlatt.html#E1:33_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="lattice4.html#T6">LATTICE4:6</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<a NAME="E6:33_1_1_1"/><i><font color="Green">E38</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="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="openlatt.html#E3:33_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="openlatt.html#E5:33_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="filter_0.html#D1">FILTER_0:def 1</a></i>;<br/>
<a NAME="E7:33_1_1_1"/>
( <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>11</sub></font> &amp; <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>12</sub></font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<a NAME="E8:33_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <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#R3">[=</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font> &amp; <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>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T27">LATTICES:27</a></i>;<br/>
<a NAME="E9:33_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <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#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="openlatt.html#E3:33_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="openlatt.html#E5:33_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<a NAME="E10:33_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <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></span>)</span> <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>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/>
<a NAME="E11:33_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <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#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E12:33_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <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="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="openlatt.html#E1:33_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="openlatt.html#E6:33_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="filter_0.html#D1">FILTER_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:33_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="openlatt.html#E7:33"><i><font color="Green">E31</font></i></a>, <a class="txt" href="openlatt.html#E11:33_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="filter_0.html#T9">FILTER_0:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:33_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="filter_0.html#K2">.)</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="filter_0.html#K2">.)</a></span> )
 
;<br/>
<a NAME="E4:33_1_1"/>
( <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> )
 
<b>by </b><i><a class="ref" href="lattice4.html#T5">LATTICE4:5</a></i>;<br/>
<a NAME="E5:33_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> )
 
<b>by </b><i><a class="txt" href="openlatt.html#E3:33_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E6:33_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> )
 
<b>by </b><i><a class="ref" href="lattice4.html#T5">LATTICE4:5</a></i>;<br/>
<a NAME="E7:33_1_1"/><b>then </b>
( <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> or <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="filter_0.html#K2">.)</a></span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="openlatt.html#K10" target="_self">SF_have</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="openlatt.html#E7:33"><i><font color="Green">E31</font></i></a>, <a class="txt" href="openlatt.html#E2:33_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="openlatt.html#E29"><i><font color="Green">Lemma23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:33_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="openlatt.html#E5:33"><i><font color="Green">E30</font></i></a>, <a class="txt" href="openlatt.html#E1:33_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="openlatt.html#E5:33_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="openlatt.html#E6:33_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:33_1"/>
( ( not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) or <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="filter_0.html#T10">FILTER_0:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:33"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="openlatt.html#E8:33"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:33"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="openlatt.html#E7:33"><i><font color="Green">E31</font></i></a></i>;<br/>


</div>
