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<span class="kw">reconsider </span><font color="Maroon" title="c2">p</font> =  <a href="polynom1.html#K8" title="POLYNOM1:func.8">0_</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>) as   <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">ALL</font> =  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>;<br/>
<span class="kw">take </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="hilb10_2.html#D6" target="_self" onmouseover="rs('hilb10_2/D6')" onmouseout="rh()">HILB10_2:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> st <br/> for <font color="Olive" title="b2">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> iff  ex <font color="Olive" title="b3">x</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ex <font color="Olive" title="b4">y</font> being   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> &amp;  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><a href="hilb10_2.html#K1" title="HILB10_2:func.1">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">p</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> iff  ex <font color="Olive" title="b2">x</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ex <font color="Olive" title="b3">y</font> being   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> &amp;  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="hilb10_2.html#K1" title="HILB10_2:func.1">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> iff  ex <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ex <font color="Olive" title="b2">y</font> being   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="hilb10_2.html#K1" title="HILB10_2:func.1">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:54_2_1"/>
( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ex <font color="Olive" title="b2">y</font> being   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="hilb10_2.html#K1" title="HILB10_2:func.1">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ex <font color="Olive" title="b2">y</font> being   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="hilb10_2.html#K1" title="HILB10_2:func.1">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) implies <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_2_1_1"/>
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ex <font color="Olive" title="b2">y</font> being   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="hilb10_2.html#K1" title="HILB10_2:func.1">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">s</font> as   <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">y</font> =  the   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ;<br/>
 <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="hilb10_2.html#K1" title="HILB10_2:func.1">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a>  the   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> </span>)</span></span>)</span>) = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#T20" onmouseover="rs('polynom2/T20')" onmouseout="rh()">POLYNOM2:20</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 

;<br/>
<span class="kw">hence </span><a NAME="E4:54_2_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ex <font color="Olive" title="b2">y</font> being   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="hilb10_2.html#K1" title="HILB10_2:func.1">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c5">x</font> being  <font color="Maroon" title="c1">n</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> , <font color="Maroon" title="c6">y</font> being   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><a NAME="E4:54_2_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="hilb10_2.html#K1" title="HILB10_2:func.1">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E5:54_2_1"/>
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_2_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="hilb10_2.html#D5" target="_self" onmouseover="rs('hilb10_2/D5')" onmouseout="rh()">Def5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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