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

<a NAME="E1:2_1_1"/><i><font color="Green" title="E1">A1</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>;<br/>
<a NAME="E2:2_1_1"/><i><font color="Green" title="E2">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> holds   <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E1:2_1_1_1"/>
<span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="conlat_1.html#R2" title="CONLAT_1:pred.2">is-SubConcept-of</a> <font color="Maroon" title="c2">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 
<b>by </b><i><a class="txt" href="conlat_2.html#E1:2_1_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="conlat_1.html#T34" title="CONLAT_1:th.34">CONLAT_1:34</a></i>;<br/>
<b>hence </b><a NAME="E2:2_1_1_1"/>
 <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#T47" title="CONLAT_1:th.47">CONLAT_1:47</a></i>; <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>
<a NAME="E3:2_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> )</span><br/>

<a NAME="E1:2_1_1_2"/>
 <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:2_1_1"><i><font color="Green" title="E2">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:2_1_1_2"/>
( <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K6" title="CONLAT_1:func.6">Concept-with-all-Attributes</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 <b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>; <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>
<a NAME="E4:2_1_1"/><b>then </b><i><font color="Green" title="E3">A3</font></i>: 
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a></i>;<br/>
<a NAME="E5:2_1_1"/><i><font color="Green" title="E4">A4</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>;<br/>
<a NAME="E6:2_1_1"/><i><font color="Green" title="E5">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</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#R3" title="LATTICES:pred.3">[=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span></span><br/>

<a NAME="E1:2_1_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="conlat_1.html#R2" title="CONLAT_1:pred.2">is-SubConcept-of</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:2_1_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="conlat_1.html#T34" title="CONLAT_1:th.34">CONLAT_1:34</a></i>;<br/>
<b>hence </b><a NAME="E2:2_1_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="conlat_1.html#T47" title="CONLAT_1:th.47">CONLAT_1:47</a></i>; <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>
<a NAME="E7:2_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span> )</span><br/>

<a NAME="E1:2_1_1_4"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E6:2_1_1"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:2_1_1_4"/>
( <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_2.html#K1" title="CONLAT_2:func.1">@</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 <b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>; <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>
<a NAME="E8:2_1_1"/><b>then </b>
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a></i>;<br/>
<b>hence </b><a NAME="E9:2_1_1"/>
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:2_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
