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<span class="kw">let </span><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> ) iff <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> )</span><br/>



<span class="kw">thus </span><a NAME="E1:58"/>
( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> implies <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> implies ( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a>
 <span class="kw">and </span><br/><a NAME="E2:58_1"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a></span><br/>

<a NAME="E3:58_1"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">L2</font>) <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">L2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="lattice2.html#T35" title="LATTICE2:th.35">LATTICE2:35</a></span>;<br/>
<a NAME="E4:58_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L1</font>) <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">L1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="lattice2.html#T35" title="LATTICE2:th.35">LATTICE2:35</a></span>;<br/>
<a NAME="E5:58_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L1</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">L2</font>)</span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">L1</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">L2</font>)</span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="filter_1.html#T30" target="_self" title="FILTER_1:th.30">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:58_1"/>
<span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T36" title="LATTICE2:th.36">LATTICE2:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:58"/>
<span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> )</span><br/>

<a NAME="E3:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>) <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T35" title="LATTICE2:th.35">LATTICE2:35</a></span>;<br/>
<a NAME="E4:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">L2</font>) <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">L2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T30" target="_self" title="FILTER_1:th.30">Th30</a></span>;<br/>
<a NAME="E5:58"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L1</font>) <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">L1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="filter_1.html#T30" target="_self" title="FILTER_1:th.30">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:58"/>
( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM2" title="LATTICES:NM.2">D_Lattice</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="lattice2.html#T36" title="LATTICE2:th.36">LATTICE2:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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