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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>, <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="E1:114"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:114"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="filter_2.html#T73" target="_self">Th73</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span> <b>by </b><i><a class="txt" href="filter_2.html#E1:114"><i><font color="Green">E50</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self">Th63</a></i>;<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E2:114_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="filter_2.html#E1:114"><i><font color="Green">E50</font></i></a>, <a class="txt" href="filter_2.html#E2:114"><i><font color="Green">E51</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self">Th63</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self">Th74</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="filter_2.html#E2:114_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/><b>hence </b><a NAME="E4:114_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E2:114_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="filter_2.html#E1:114"><i><font color="Green">E50</font></i></a>, <a class="txt" href="filter_2.html#E2:114"><i><font color="Green">E51</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self">Th63</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self">Th74</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="filter_2.html#E2:114_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>
;<br/><b>hence </b><a NAME="E4:114_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:114"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span> is  <a href="lattices.html#V13">lower-bounded</a> <a href="lattices.html#V14">upper-bounded</a> <a href="lattices.html#NM1">Lattice</a>
 
<b>by </b><i><a class="txt" href="filter_2.html#E4:114"><i><font color="Green">E52</font></i></a>, <a class="ref" href="lattices.html#D13">LATTICES:def 13</a>, <a class="ref" href="lattices.html#D14">LATTICES:def 14</a></i>;<br/>
<b>hence </b><a NAME="E7:114"/>
 <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span> is <a href="lattices.html#V15">bounded</a>
 ;<br/>

<b>thus </b><a NAME="E8:114"/>
(  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="filter_2.html#E4:114"><i><font color="Green">E52</font></i></a>, <a class="txt" href="filter_2.html#E5:114"><i><font color="Green">E53</font></i></a>, <a class="txt" href="filter_2.html#E6:114"><i><font color="Green">E54</font></i></a>, <a class="ref" href="lattices.html#D16">LATTICES:def 16</a>, <a class="ref" href="lattices.html#D17">LATTICES:def 17</a></i>;<br/>


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