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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice4.html#NM2" title="LATTICE4:NM.2">ClosedSubset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>  holds <br/> <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">P</font>) is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice4.html#NM2" title="LATTICE4:NM.2">ClosedSubset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>  implies  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">P</font>) is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:98"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">c</font></span>)</span>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattices.html#D11" onmouseover="rs('lattices/D11')" onmouseout="rh()">LATTICES:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">P</font>) is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">a9</font>, <font color="Maroon" title="c4">b9</font>, <font color="Maroon" title="c5">c9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">P</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattices.html#D11" onmouseover="rs('lattices/D11')" onmouseout="rh()">LATTICES:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b9</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">c9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b9</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c9</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c3">a9</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c4">b9</font>, <font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c5">c9</font>, <font color="Maroon" title="c9">bc</font> = <font color="Maroon" title="c4">b9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">c9</font>, <font color="Maroon" title="c10">ab</font> = <font color="Maroon" title="c3">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b9</font>, <font color="Maroon" title="c11">ac</font> = <font color="Maroon" title="c3">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T68" target="_self" onmouseover="rs('filter_2/T68')" onmouseout="rh()">Th68</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">c9</font></span>)</span> = 
<font color="Maroon" title="c6">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">bc</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" onmouseover="rs('filter_2/T73')" onmouseout="rh()">Th73</a></span>
<br/>.= 
<font color="Maroon" title="c6">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" onmouseover="rs('filter_2/T73')" onmouseout="rh()">Th73</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E42">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">ab</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" onmouseover="rs('filter_2/T73')" onmouseout="rh()">Th73</a></span>
<br/>.= 
<font color="Maroon" title="c10">ab</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c11">ac</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" onmouseover="rs('filter_2/T73')" onmouseout="rh()">Th73</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b9</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" onmouseover="rs('filter_2/T73')" onmouseout="rh()">Th73</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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