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<b>consider </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="zfmisc_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lattices.html#G3" target="_self">LattStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> #) as  <a href="lattices.html#V3" target="_self">strict</a> <a href="lattices.html#NM1">Lattice</a> <b>by </b><i><a class="txt" href="lattices.html#E18"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<a NAME="E3:43_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="lattices.html#V16" target="_self">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19" target="_self">LATTICES:def 19</a><br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E1:43_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6" target="_self">Top</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6" target="_self">Top</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="lattices.html#E4"><i><font color="Green">Lemma2</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18" target="_self">LATTICES:def 18</a><br/>

<b>thus </b><a NAME="E2:43_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5" target="_self">Bottom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5" target="_self">Bottom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="lattices.html#E4"><i><font color="Green">Lemma2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:43_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> is   <a href="lattices.html#NM4">0_Lattice</a> &amp; <font color="Maroon">c<sub>3</sub></font> is   <a href="lattices.html#NM5">1_Lattice</a> )
 
<b>by </b><i><a class="txt" href="lattices.html#E19"><i><font color="Green">Lemma17</font></i></a>, <a class="txt" href="lattices.html#E20"><i><font color="Green">Lemma18</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="lattices.html#NM7">C_Lattice</a> <b>by </b><i><a class="txt" href="lattices.html#E3:43_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lattices.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E6:43_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattices.html#E4"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E7:43_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="lattices.html#V11" target="_self">distributive</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E8:43_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="lattices.html#V17" target="_self">Boolean</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D20" target="_self">Def20</a></i>;<br/>
<b>hence </b><a NAME="E9:43_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="lattices.html#NM1">Lattice</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="lattices.html#V17" target="_self">Boolean</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="lattices.html#V3" target="_self">strict</a> )
 ;<br/>


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