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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">CP1</font>, <font color="Olive" title="b2">CP2</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/> ( <font color="Olive" title="b1">CP1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">CP2</font> iff <font color="Olive" title="b1">CP1</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> <font color="Olive" title="b2">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">CP1</font>, <font color="Maroon" title="c3">CP2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">CP1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">CP2</font> iff <font color="Maroon" title="c2">CP1</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> <font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">CL</font> =  <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font>;<br/>
<div><span class="lab"><font color="Green" title="E49">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:93_1"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<font color="Maroon" title="c2">CP1</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> <font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">CP1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">CP2</font></span><br/><a NAME="E2:93_1"/><span class="kw">then </span>
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#T31" target="_self" title="CONLAT_1:th.31">Th31</a></span>;<br/><a NAME="E3:93_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A3</font></span>: 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">O</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c6">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E5:93_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K10" title="CONLAT_1:func.10">B-join</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">O</font>,<font color="Maroon" title="c6">A</font> #)
 <span class="kw">and </span><br/><a NAME="E6:93_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon" title="c5">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:93_1"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#D22" target="_self" title="CONLAT_1:def.22">Def22</a></span>;<br/><a NAME="E8:93_1"/>
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="conlat_1.html#D19" target="_self" title="CONLAT_1:def.19">Def19</a></span>;<br/><a NAME="E9:93_1"/><span class="kw">then </span>
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">O</font> = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:93_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="conlat_1.html#D13" target="_self" title="CONLAT_1:def.13">Def13</a></span>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#D13" target="_self" title="CONLAT_1:def.13">Def13</a></span>
;<br/><a NAME="E11:93_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">CP1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">CP2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CP2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E5:93_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E7:93_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:93_1"/>
<font color="Maroon" title="c2">CP1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">CP2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:93_2"/>
<font color="Maroon" title="c2">CP1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">CP2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">CP1</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> <font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span><br/><a NAME="E2:93_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">CP1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">CP2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CP2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>;<br/><a NAME="E3:93_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A7</font></span>: 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</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 href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">CP1</font>,<font color="Maroon" title="c3">CP2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CP2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>;<br/><a NAME="E4:93_2"/>
 ex <font color="Olive" title="b1">O</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <span class="p1">(<span class="default"><a href="conlat_1.html#K10" title="CONLAT_1:func.10">B-join</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) &amp; <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#D22" target="_self" title="CONLAT_1:def.22">Def22</a></span>;<br/><a NAME="E5:93_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_2"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:93_2"/><span class="kw">then </span>
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">CP1</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E7:93_2"/>
<font color="Maroon" title="c2">CP1</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> <font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#T31" target="_self" title="CONLAT_1:th.31">Th31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:93"/>
( <font color="Maroon" title="c2">CP1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">CP2</font> iff <font color="Maroon" title="c2">CP1</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> <font color="Maroon" title="c3">CP2</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E49">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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