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<div class="add">

<b>thus </b><a NAME="E1:71_1_1"/>
 <a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> &amp;  <a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> )</span><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" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span>) st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">b</font></span><br/>

<b>take </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">b</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c5">b'</font> = <font color="Maroon" title="c2">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="robbins1.html#D11" target="_self" title="ROBBINS1:def.11">Def11</a></i>;<br/>
<b>thus </b><a NAME="E2:71_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="robbins1.html#D8" target="_self" title="ROBBINS1:def.8">Def8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E3:71_1_1_1"/>
<font color="Maroon" title="c2">b</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="robbins1.html#D8" target="_self" title="ROBBINS1:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<i><font color="Green" title="E39">A1</font></i>: <font color="Maroon" title="c4">a'</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  = 
<font color="Maroon" title="c3">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="ref" href="robbins1.html#D11" target="_self" title="ROBBINS1:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c5">b'</font>
<b>by </b><i><a class="ref" href="robbins1.html#T3" target="_self" title="ROBBINS1:th.3">Th3</a></i>
;<br/>
<b>thus </b><font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">b</font> = 
<font color="Maroon" title="c4">a'</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <font color="Maroon" title="c5">b'</font>
<b>by </b><i><a class="ref" href="robbins1.html#D11" target="_self" title="ROBBINS1:def.11">Def11</a></i>
<br/>.= 
 <a href="robbins1.html#K7" title="ROBBINS1:func.7">Bot</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="robbins1.html#E4:71_1_1_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="robbins1.html#T9" target="_self" title="ROBBINS1:th.9">Th9</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T36" target="_self" title="ROBBINS1:th.36">Th36</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span><br/>

<b>hence </b><a NAME="E6:71_1_1_1"/>
<font color="Maroon" title="c2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</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" title="LATTICES:def.11">LATTICES:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a></span>
<div class="add"><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">b'</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c7">c'</font> = <font color="Maroon" title="c4">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="robbins1.html#D11" target="_self" title="ROBBINS1:def.11">Def11</a></i>;<br/><a NAME="E2:71_1_1_2"/><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c6">b'</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c7">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <font color="Maroon" title="c7">c'</font> )
 <b>by </b><i><a class="ref" href="robbins1.html#D11" target="_self" title="ROBBINS1:def.11">Def11</a></i>;<br/><b>hence </b><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span> = 
<font color="Maroon" title="c5">a'</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b'</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c7">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#D11" target="_self" title="ROBBINS1:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a'</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <font color="Maroon" title="c6">b'</font></span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a'</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <font color="Maroon" title="c7">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T31" target="_self" title="ROBBINS1:th.31">Th31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="txt" href="robbins1.html#E2:71_1_1_2"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="robbins1.html#D11" target="_self" title="ROBBINS1:def.11">Def11</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:71_1_1"/>
 <a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a></span><br/>

<b>thus </b><a NAME="E4:71_1_1"/>
 <a href="robbins1.html#K8" title="ROBBINS1:func.8">CLatt</a> <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
