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<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>assume </b><a NAME="E1:63"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM5">1_Lattice</a> &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="lattices.html#NM5">1_Lattice</a> )
 ;<br/>

<a NAME="E2:63"/><b>then </b><i><font color="Green">E43</font></i>: 
<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#V14">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="filter_1.html#T41" target="_self">Th41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </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 <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>;<br/><b>consider </b><font color="Maroon">c<sub>4</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>5</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="E2:63_1"/><i><font color="Green">E44</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>4</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="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>thus </b><span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> = 
<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"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</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#E2:63_1"><i><font color="Green">E44</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"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</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#E1:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="lattices.html#T44">LATTICES:44</a></i>
<br/>.= 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</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#E1:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="lattices.html#T44">LATTICES:44</a></i>
;<br/><b>hence </b><a NAME="E4:63_1"/>
<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"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</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"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
 <a href="lattices.html#K6">Top</a> <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> <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"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</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#E2:63"><i><font color="Green">E43</font></i></a>, <a class="ref" href="lattices.html#D17">LATTICES:def 17</a></i>;<br/>


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