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<a NAME="E1:71_1_1"/><i><font color="Green">E40</font></i>: 
the <a href="robbins1.html#U1" target="_self">Compl</a> of <span class="p1">(<span class="default"><a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R1">=</a> the <a href="robbins1.html#U1" target="_self">Compl</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="robbins1.html#D11" target="_self">Def11</a></i>;<br/>
<b>thus </b><a NAME="E2:71_1_1"/>
 <a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V16">complemented</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_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 <span class="p1">(<span class="default"><a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a><br/>

<b>take </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="robbins1.html#K3" target="_self">`</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="robbins1.html#D11" target="_self">Def11</a></i>;<br/>
<b>thus </b><a NAME="E2:71_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="robbins1.html#D8" target="_self">Def8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a><br/>

<b>thus </b><a NAME="E3:71_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="robbins1.html#D8" target="_self">Def8</a></i>;<br/>

<i><font color="Green">E41</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a>  = 
the <a href="robbins1.html#U1" target="_self">Compl</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> 
<b>by </b><i><a class="txt" href="robbins1.html#E1:71_1_1"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="robbins1.html#T3" target="_self">Th3</a></i>
;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="robbins1.html#T35" target="_self">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 

<br/>.= 
 <a href="robbins1.html#K7" target="_self">Bot</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="robbins1.html#E4:71_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="robbins1.html#T9" target="_self">Th9</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T36" target="_self">Th36</a></i>
;<br/>

<b>hence </b><a NAME="E6:71_1_1_1"/>
<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> <span class="p1">(<span class="default"><a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="robbins1.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E2:71_1_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</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> <font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="robbins1.html#T35" target="_self">Th35</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T35" target="_self">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T31" target="_self">Th31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="robbins1.html#E2:71_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="robbins1.html#T35" target="_self">Th35</a></i>
;<br/><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E4:71_1_1"/>
 <a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V13">lower-bounded</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15">LATTICES:def 15</a><br/>

<b>thus </b><a NAME="E5:71_1_1"/>
 <a href="robbins1.html#K8" target="_self">CLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V14">upper-bounded</a>
 ;<br/>


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