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<div class="add">

<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>thus </b><a NAME="E1:58"/>
( <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM2">D_Lattice</a> &amp; <font color="Maroon">c<sub>2</sub></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">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#NM2">D_Lattice</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1"/>
( <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM2">D_Lattice</a> &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="lattices.html#NM2">D_Lattice</a> )
 ;<br/>

<a NAME="E2:58_1"/><b>then </b>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) )
 
<b>by </b><i><a class="ref" href="lattice2.html#T35">LATTICE2:35</a></i>;<br/>
<a NAME="E3:58_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)</span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#R6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>)</span><a href="filter_1.html#K6" target="_self">:|</a></span>
 
<b>by </b><i><a class="ref" href="filter_1.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E4:58_1"/>
<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#NM2">D_Lattice</a>
 <b>by </b><i><a class="ref" href="lattice2.html#T36">LATTICE2:36</a></i>;<br/>


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

<b>assume </b><a NAME="E2:58"/>
<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#NM2">D_Lattice</a>
 ;<br/>

<a NAME="E3:58"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>(<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="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">H<sub>3</sub></font>(<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>)
 
<b>by </b><i><a class="ref" href="lattice2.html#T35">LATTICE2:35</a></i>;<br/>
<a NAME="E4:58"/><b>then </b>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) )
 
<b>by </b><i><a class="ref" href="filter_1.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E5:58"/>
( <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM2">D_Lattice</a> &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="lattices.html#NM2">D_Lattice</a> )
 <b>by </b><i><a class="ref" href="lattice2.html#T36">LATTICE2:36</a></i>;<br/>


</div>
