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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>  <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</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">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  ) ) implies ( <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a>  iff for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st not <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/> ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</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="c1">L</font> is  <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a>  iff for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st not <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/> ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> ) )</span><br/>

<div><span class="kw">hereby </span> <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">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st not <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/> ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> ) ) implies <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</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">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st not <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/> ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> implies  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> ) )</span><br/><span class="kw">assume </span><a NAME="E2:42_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
not <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )</span><br/><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/><a NAME="E4:42_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="waybel_3.html#D5" target="_self" title="WAYBEL_3:def.5">Def5</a></span>;<br/><a NAME="E5:42_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></span>;<br/><a NAME="E6:42_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font> implies <font color="Maroon" title="c3">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E8:42_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">and </span><br/><a NAME="E9:42_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
not <font color="Maroon" title="c4">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c5">u</font> = <font color="Maroon" title="c4">u</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="c5">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )</span><br/><span class="kw">thus </span><a NAME="E10:42_1"/>
( <font color="Maroon" title="c5">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="waybel_3.html#T7" target="_self" title="WAYBEL_3:th.7">Th7</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">assume </span><a NAME="E3:42"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st not <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/> ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</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="c1">L</font> is  <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_3.html#D5" target="_self" title="WAYBEL_3:def.5">WAYBEL_3:def 5</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span></span><br/>

<a NAME="E4:42"/>
( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/>
<a NAME="E5:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A9</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></span>;<br/>
<a NAME="E6:42"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T9" target="_self" title="WAYBEL_3:th.9">Th9</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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="c3">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font> implies <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">and </span><br/><a NAME="E2:42_2"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
not <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c4">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:42_2"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">and </span><br/><a NAME="E5:42_2"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
not <font color="Maroon" title="c4">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E2:42_2"><span class="lab"><font color="Green" title="E23">A12</font></span></a></span>;<br/><a NAME="E6:42_2"/>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E24">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:42_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E5:42_2"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</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="E8:42"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</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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