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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V3" title="YELLOW_0:attr.3">bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:143_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a> )
 ;<br/><a NAME="E2:143_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">x'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>for <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b3">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">x'</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">x'</font> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E118"><i><font color="Green" title="E81">Lm6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:143_1"/>
( <font color="Maroon" title="c1">L</font> <a href="waybel_1.html#NR13" title="WAYBEL_1:NR.13">is_a_Heyting_algebra</a>  &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="waybel_1.html#E119"><i><font color="Green" title="E82">Lm7</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:143"/><i><font color="Green" title="E85">A1</font></i>: 
<font color="Maroon" title="c1">L</font> <a href="waybel_1.html#NR13" title="WAYBEL_1:NR.13">is_a_Heyting_algebra</a> 
 <b>and </b><br/><a NAME="E3:143"/><i><font color="Green" title="E86">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<b>thus </b><a NAME="E4:143"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:143"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="waybel_1.html#T69" target="_self" title="WAYBEL_1:th.69">Th69</a></i>;<br/>

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D24" target="_self" title="WAYBEL_1:def.24">WAYBEL_1:def 24</a><br/>

<b>take </b>
 <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font>
;<br/>

<b>thus </b><a NAME="E5:143"/>
 <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:143"><i><font color="Green" title="E85">A1</font></i></a>, <a class="txt" href="waybel_1.html#E3:143"><i><font color="Green" title="E86">A2</font></i></a>, <a class="ref" href="waybel_1.html#T89" target="_self" title="WAYBEL_1:th.89">Th89</a></i>;<br/>


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