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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font> : ( not  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) is <a href="conlat_1.html#V7" target="_self">empty</a> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/>
<a NAME="E1:66_1_1"/>
not <span class="p1">{<span class="default"> <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font> : ( not  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) is <a href="conlat_1.html#V7" target="_self">empty</a> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:66_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:66_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<a NAME="E6:66_1_1_1"/><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="conlat_1.html#E3:66_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="conlat_1.html#E5:66_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="conlat_1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E7:66_1_1_1"/><b>then </b>
not  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="conlat_1.html#V7" target="_self">empty</a>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E32"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<a NAME="E8:66_1_1_1"/><b>then </b>
 <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font> : ( not  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) is <a href="conlat_1.html#V7" target="_self">empty</a> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="conlat_1.html#E6:66_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:66_1_1_1"/>
not <span class="p1">{<span class="default"> <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font> : ( not  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) is <a href="conlat_1.html#V7" target="_self">empty</a> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font> : ( not  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) is <a href="conlat_1.html#V7" target="_self">empty</a> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E3:66_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="conlat_1.html#V6" target="_self">strict</a> non <a href="conlat_1.html#V7" target="_self">empty</a>  <a href="conlat_1.html#L3" target="_self">ConceptStr</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="conlat_1.html#V6" target="_self">strict</a> non <a href="conlat_1.html#V7" target="_self">empty</a>  <a href="conlat_1.html#L3" target="_self">ConceptStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:66_1_1_2_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> #) &amp; not  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> #) is <a href="conlat_1.html#V7" target="_self">empty</a> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:66_1_1_2_1"/>
( <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E3:66_1_1_2_1"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:66_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <font color="Maroon">c<sub>4</sub></font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:66_1_1"/>
<span class="p1">{<span class="default"> <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font> : ( not  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> #) is <a href="conlat_1.html#V7" target="_self">empty</a> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 ;<br/>


</div>
