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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="filter_0.html#NM2">I_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>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:57"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</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>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 ;<br/>

<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>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> = 
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
<b>by </b><i><a class="ref" href="filter_0.html#T47" target="_self">Th47</a></i>
<br/>.= 
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="filter_0.html#K3" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
<b>by </b><i><a class="ref" href="filter_0.html#T47" target="_self">Th47</a></i>
<br/>.= 
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
<b>by </b><i><a class="ref" href="filter_0.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="filter_0.html#K6" target="_self">"/\"</a> <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span>
<b>by </b><i><a class="ref" href="filter_0.html#T51" target="_self">Th51</a></i>
;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <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="E4:57"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</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"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span> )
 <b>by </b><i><a class="txt" href="filter_0.html#E1:57"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E5:57"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E4:57"><i><font color="Green">E41</font></i></a>, <a class="ref" href="filter_0.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E6:57"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <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>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <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>5</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T27">LATTICES:27</a></i>;<br/>
<a NAME="E7:57"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> <a href="filter_0.html#K4" target="_self">=&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E4:57"><i><font color="Green">E41</font></i></a>, <a class="ref" href="filter_0.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E8:57"/>
<font color="Maroon">c<sub>3</sub></font> <a href="filter_0.html#K4" target="_self">=&gt;</a> <font color="Maroon">c<sub>2</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"><font color="Maroon">c<sub>4</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 <b>by </b><i><a class="txt" href="filter_0.html#E4:57"><i><font color="Green">E41</font></i></a>, <a class="ref" href="filter_0.html#T9" target="_self">Th9</a></i>;<br/>


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