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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattices.html#V4">join-commutative</a> <a href="lattices.html#V6">meet-commutative</a> <a href="lattices.html#V11">distributive</a> <a href="robbins1.html#V6">join-idempotent</a> <a href="sheffer1.html#V1" target="_self">upper-bounded'</a> <a href="sheffer1.html#V2" target="_self">lower-bounded'</a> <a href="sheffer1.html#V3" target="_self">distributive'</a> <a href="sheffer1.html#V4" target="_self">complemented'</a>  <a href="lattices.html#L3">LattStr</a> ;<br/>

<a NAME="E1:33"/>
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 <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<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>;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:33_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="sheffer1.html#R1" target="_self">is_a_complement'_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="sheffer1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:33_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="sheffer1.html#E2:33_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="sheffer1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E4:33_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="sheffer1.html#T19" target="_self">Th19</a>, <a class="ref" href="sheffer1.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E5:33_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>
<b>hence </b><a NAME="E6:33_1"/>
ex <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> st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V16">complemented</a>
 <b>by </b><i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>


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