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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V3">bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:143_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2" target="_self">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V10" target="_self">complemented</a> )
 ;<br/><a NAME="E2:143_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E118"><i><font color="Green">Lemma83</font></i></a></i>;<br/><b>hence </b><a NAME="E3:143_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="waybel_1.html#NR13" target="_self">is_a_Heyting_algebra</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="waybel_1.html#E119"><i><font color="Green">Lemma84</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">E87</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="waybel_1.html#NR13" target="_self">is_a_Heyting_algebra</a> 
 <b>and </b><br/><a NAME="E3:143"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>thus </b><a NAME="E4:143"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2" target="_self">distributive</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:143"><i><font color="Green">E87</font></i></a>, <a class="ref" href="waybel_1.html#T69" target="_self">Th69</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D24" target="_self">WAYBEL_1:def 24</a><br/>

<b>take </b>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E5:143"/>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#R6" target="_self">is_a_complement_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:143"><i><font color="Green">E87</font></i></a>, <a class="txt" href="waybel_1.html#E3:143"><i><font color="Green">E88</font></i></a>, <a class="ref" href="waybel_1.html#T89" target="_self">Th89</a></i>;<br/>


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