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<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> 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="E1:62"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V14">upper-bounded</a>
 ;<br/>

<a NAME="E2:62"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V13">lower-bounded</a> &amp;  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice2.html#T64">LATTICE2:64</a>, <a class="ref" href="lattice2.html#T79">LATTICE2:79</a></i>;<br/>
<b>assume </b><a NAME="E3:62"/>
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:62"/><b>then </b>
( <span class="p1"><a href="filter_2.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="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="filter_2.html#R1" target="_self">=</a> <span class="p1"><a href="filter_2.html#K1" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span></span><a href="filter_2.html#K1" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_2.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="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> ) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <span class="p1"><a href="filter_2.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="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E2:62"><i><font color="Green">E25</font></i></a>, <a class="ref" href="filter_2.html#T37" target="_self">Th37</a>, <a class="ref" href="filter_0.html#T31">FILTER_0:31</a></i>;<br/>
<b>hence </b><a NAME="E5:62"/>
( <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="filter_2.html#R1" target="_self">=</a> <span class="p1"><a href="filter_2.html#K17" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="filter_2.html#K17" target="_self">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


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