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

<b>let </b><font color="Maroon">L1</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">L2</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>



<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon">L1</font> is <a href="lattices.html#V15">bounded</a> &amp; <font color="Maroon">L2</font> is <a href="lattices.html#V15">bounded</a> implies <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is <a href="lattices.html#V15">bounded</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/>
( <font color="Maroon">L1</font> is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L1</font> is <a href="lattices.html#V14">upper-bounded</a> &amp; <font color="Maroon">L2</font> is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L2</font> is <a href="lattices.html#V14">upper-bounded</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15">LATTICES:def 15</a><br/>

<b>hence </b><a NAME="E2:61_1"/>
( <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is <a href="lattices.html#V13">lower-bounded</a> &amp; <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</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>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15">LATTICES:def 15</a><br/>


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

<b>assume </b><a NAME="E2:61"/>
( <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is <a href="lattices.html#V13">lower-bounded</a> &amp; <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is <a href="lattices.html#V14">upper-bounded</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15">LATTICES:def 15</a><br/>

<b>hence </b><a NAME="E3:61"/>
( <font color="Maroon">L1</font> is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L1</font> is <a href="lattices.html#V14">upper-bounded</a> &amp; <font color="Maroon">L2</font> is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L2</font> is <a href="lattices.html#V14">upper-bounded</a> )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15">LATTICES:def 15</a><br/>


</div>
