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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="robbins1.html#L2">ComplLattStr</a> ;<br/>

<b>assume </b><a NAME="E1:47"/><i><font color="Green">E22</font></i>: 
(  <a href="robbins1.html#G2">ComplLattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="robbins1.html#G2">ComplLattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">c<sub>2</sub></font> #) &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="robbins3.html#V9" target="_self">with_Top</a> )
 ;<br/>

<a NAME="E2:47"/><i><font color="Green">E23</font></i>: 
 <a href="lattices.html#G2">\/-SemiLattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="lattices.html#G2">\/-SemiLattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>2</sub></font> #)
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:47"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:47"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><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 <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</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="txt" href="robbins3.html#E1:47"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E2:47_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="robbins1.html#K3">`</a>  )
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:47"><i><font color="Green">E22</font></i></a>, <a class="ref" href="robbins3.html#T18" target="_self">Th18</a></i>;<br/>
<font color="Maroon">c<sub>3</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E2:47"><i><font color="Green">E23</font></i></a>, <a class="txt" href="robbins3.html#E2:47_1"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E1:47"><i><font color="Green">E22</font></i></a>, <a class="ref" href="robbins3.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E2:47"><i><font color="Green">E23</font></i></a>, <a class="txt" href="robbins3.html#E2:47_1"><i><font color="Green">E24</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:47_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="robbins3.html#V9" target="_self">with_Top</a>
 <b>by </b><i><a class="ref" href="robbins3.html#D7" target="_self">Def7</a></i>;<br/>


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