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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="conlat_1.html#NM1">FormalContext</a>;<br/>

<a NAME="E1:9"/>
 <a href="lattice3.html#K15">"\/"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#T51">LATTICE3:51</a></i>;<br/>
<a NAME="E2:9"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E3:9"/><b>then </b><i><font color="Green">E5</font></i>: 
 <a href="conlat_2.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E4:9"/><i><font color="Green">E6</font></i>: 
 <a href="conlat_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#R3">is_less_than</a>  <a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:9_1"/><i><font color="Green">E7</font></i>: 
 <a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font> <a href="conlat_1.html#R2">is-SubConcept-of</a> <font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#T34">CONLAT_1:34</a></i>;<br/>
 <a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="conlat_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="conlat_2.html#K1" target="_self">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="conlat_1.html#K12">@</a> 
<b>by </b><i><a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
;<br/>
<b>hence </b><a NAME="E4:9_1"/>
 <a href="conlat_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:9_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="conlat_1.html#T47">CONLAT_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:9"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3">is_less_than</a>  <a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="conlat_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><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="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:9_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R3">is_less_than</a>  <a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:9_2"/>
 <a href="conlat_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E3:9_2"/><b>then </b>
 <a href="conlat_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:9_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a>  <a href="conlat_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="conlat_2.html#E1:9_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:9"/><b>then </b>
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="conlat_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E4:9"><i><font color="Green">E6</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E7:9"/><b>then </b>
 <a href="conlat_2.html#K2" target="_self">"/\"</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="conlat_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:9"/>
(  <a href="conlat_2.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#K5">Concept-with-all-Objects</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="conlat_2.html#K2" target="_self">"/\"</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#K6">Concept-with-all-Attributes</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:9"><i><font color="Green">E5</font></i></a>, <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>


</div>
