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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">KX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">i</font> being   <a href="int_1.html#V2" title="INT_1:attr.2">dim-like</a>   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">number</a>   st  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp;  <a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Olive" title="b1">KX</font>,<font color="Olive" title="b2">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">KX</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">KX</font> #) holds <br/> <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b1">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">KX</font> be    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</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">i</font> being   <a href="int_1.html#V2" title="INT_1:attr.2">dim-like</a>   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">number</a>   st  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp;  <a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">KX</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> #) holds <br/> <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font> be   <a href="int_1.html#V2" title="INT_1:attr.2">dim-like</a>   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">number</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"> (  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp;  <a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">KX</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> #) implies  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:108"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 <a href="simplex0.html#K9" title="SIMPLEX0:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</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"> (  not  <a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">KX</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> #) or  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:108_1"/>
( <font color="Maroon" title="c2">KX</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  or <font color="Maroon" title="c2">KX</font> is  <a href="simplex0.html#NV9" title="SIMPLEX0:NV.9">with_non-empty_element</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_1"/>
<font color="Maroon" title="c2">KX</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</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"> (  not  <a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">KX</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> #) or  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:108_1_1"/>
(  not  <a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">KX</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> #) or  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="simplex0.html#T22" target="_self" onmouseover="rs('simplex0/T22')" onmouseout="rh()">Th22</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c2">KX</font> is  <a href="simplex0.html#NV9" title="SIMPLEX0:NV.9">with_non-empty_element</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"> (  not  <a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">KX</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> #) or  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c4">i1</font> = <font color="Maroon" title="c3">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><span class="kw">assume </span><a NAME="E3:108_1_2"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 <a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">KX</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</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">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font></span><br/><div><span class="lab"><font color="Green" title="E44">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="108_1_2_1"><span class="kw">now </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">  for <font color="Olive" title="b1">S</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">KX</font>  st <font color="Olive" title="b1">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">KX</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"> ( <font color="Maroon" title="c5">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font> )</span><br/><span class="kw">assume </span><a NAME="E1:108_1_2_1"/>
<font color="Maroon" title="c5">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font></span><br/><a NAME="E2:108_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="simplex0.html#K2" title="SIMPLEX0:func.2">subset-closed_closure_of</a> <span class="p1">(<span class="default"><a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> (<font color="Maroon" title="c4">i1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2"><span class="lab"><font color="Green" title="E43">A3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:108_1_2_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
( <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> (<font color="Maroon" title="c4">i1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T2" target="_self" onmouseover="rs('simplex0/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E5:108_1_2_1"/>
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">y</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">i1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108_1_2_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="simplex0.html#D2" target="_self" onmouseover="rs('simplex0/D2')" onmouseout="rh()">Def2</a>, <a class="ref" href="card_1.html#T11" onmouseover="rs('card_1/T11')" onmouseout="rh()">CARD_1:11</a></span>;<br/><a NAME="E6:108_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A6</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c4">i1</font></span>)</span>
 ;<br/><a NAME="E7:108_1_2_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font></span>)</span></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E8:108_1_2_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108_1_2_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="nat_1.html#T40" onmouseover="rs('nat_1/T40')" onmouseout="rh()">NAT_1:40</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/></div><span class="kw">end;</span></div><a NAME="E5:108_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> holds <br/><font color="Olive" title="b1">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2"><span class="lab"><font color="Green" title="E43">A3</font></span></a></span>;<br/><a NAME="E6:108_1_2"/><span class="kw">then </span>
 <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c2">KX</font> is  <a href="finset_1.html#V5" title="FINSET_1:attr.5">finite-membered</a> 
 ;<br/><a NAME="E7:108_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">KX</font> is  <a href="matroid0.html#V3" title="MATROID0:attr.3">finite-membered</a> 
 ;<br/><span class="kw">hence </span><a NAME="E8:108_1_2"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1_2"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E4:108_1_2"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="simplex0.html#T25" target="_self" onmouseover="rs('simplex0/T25')" onmouseout="rh()">Th25</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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