<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  iff <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  )</span><br/>

<div><span class="lab"><font color="Green" title="E16">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st <font color="Olive" title="b1">L</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  holds <br/><font color="Olive" title="b1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  implies <font color="Maroon" title="c2">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:21_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c2">L</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> </span><br/><span class="kw">thus </span><a NAME="E2:21_1"/>
<font color="Maroon" title="c2">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D12" onmouseover="rs('lattice3/D12')" onmouseout="rh()">LATTICE3:def 12</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span> st <br/>( <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span>  holds <br/>(  not <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">Y</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> </span> } </span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E2:21_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> </span> } </span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <span class="p1"> { <span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Olive" title="b2">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> </span> } </span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c5">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">a</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">x</font> &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span>  holds <br/>(  not <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:21_1_1"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">x</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span>  holds <br/>(  not <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D9" onmouseover="rs('lattice3/D9')" onmouseout="rh()">LATTICE3:def 9</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E2:21_1_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> </span> } </span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D9" onmouseover="rs('lattice3/D9')" onmouseout="rh()">LATTICE3:def 9</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> </span> } </span>  or <font color="Maroon" title="c8">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">y</font></span><br/>

<a NAME="E2:21_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Olive" title="b1">z</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:21_1_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:21_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T2" target="_self" onmouseover="rs('yellow_7/T2')" onmouseout="rh()">Th2</a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:21_1_1"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font></span><br/>

<a NAME="E6:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T9" target="_self" onmouseover="rs('yellow_7/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<a NAME="E7:21_1_1"/><span class="kw">then </span>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> </span> } </span> 
 
;<br/>
<a NAME="E8:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/>
<a NAME="E9:21_1_1"/>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E10:21_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="yellow_7.html#T1" target="_self" onmouseover="rs('yellow_7/T1')" onmouseout="rh()">Th1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E2:21"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  implies <font color="Maroon" title="c1">L</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T3" onmouseover="rs('yellow_0/T3')" onmouseout="rh()">YELLOW_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21"/>
( <font color="Maroon" title="c1">L</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  iff <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
