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<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/>



<a NAME="E1:66"/>
( ( <font color="Maroon">L1</font> is   <a href="lattices.html#NM8">B_Lattice</a> implies ( <font color="Maroon">L1</font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">L1</font> is   <a href="lattices.html#NM2">D_Lattice</a> ) ) &amp; ( <font color="Maroon">L1</font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">L1</font> is   <a href="lattices.html#NM2">D_Lattice</a> implies <font color="Maroon">L1</font> is   <a href="lattices.html#NM8">B_Lattice</a> ) &amp; ( <font color="Maroon">L2</font> is   <a href="lattices.html#NM8">B_Lattice</a> implies ( <font color="Maroon">L2</font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM2">D_Lattice</a> ) ) &amp; ( <font color="Maroon">L2</font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM2">D_Lattice</a> implies <font color="Maroon">L2</font> is   <a href="lattices.html#NM8">B_Lattice</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#NM7">C_Lattice</a> implies ( <font color="Maroon">L1</font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM7">C_Lattice</a> ) ) &amp; ( <font color="Maroon">L1</font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM7">C_Lattice</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#NM7">C_Lattice</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#NM2">D_Lattice</a> implies ( <font color="Maroon">L1</font> is   <a href="lattices.html#NM2">D_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM2">D_Lattice</a> ) ) &amp; ( <font color="Maroon">L1</font> is   <a href="lattices.html#NM2">D_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM2">D_Lattice</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#NM2">D_Lattice</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#NM8">B_Lattice</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#NM7">C_Lattice</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#NM2">D_Lattice</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#NM7">C_Lattice</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#NM2">D_Lattice</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#NM8">B_Lattice</a> ) )
 
<b>by </b><i>, , <a class="ref" href="lattices.html#D20">LATTICES:def 20</a></i>;<br/>
<b>hence </b><a NAME="E2:66"/>
( ( <font color="Maroon">L1</font> is   <a href="lattices.html#NM8">B_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM8">B_Lattice</a> ) iff <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#NM8">B_Lattice</a> )
 ;<br/>


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