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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font> holds   <a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a></span><br/>



<a NAME="E1:17"/>
(  <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">Y</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</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 <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> #) &amp;  <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="c3">Z</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">Z</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 <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">Z</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">Z</font></span>)</span> #) )
 
<b>by </b><i><a class="ref" href="waybel25.html#D2" title="WAYBEL25:def.2">WAYBEL25:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">f'</font> = <font color="Maroon" title="c4">f</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">Z</font></span>)</span> <b>by </b><i><a class="ref" href="yellow12.html#T36" title="YELLOW12:th.36">YELLOW12:36</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">Xf</font> =  <a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font>;<br/>
<b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> or <span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">g2</font> = <font color="Maroon" title="c8">b</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T1" target="_self" title="WAYBEL26:th.1">Th1</a></i>;<br/>
<b>assume </b><a NAME="E4:17"/>
<font color="Maroon" title="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">b</font>
 ; <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="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span><br/>

<a NAME="E5:17"/><b>then </b><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c9">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">g2</font>
 
<b>by </b><i><a class="ref" href="waybel26.html#T3" target="_self" title="WAYBEL26:th.3">Th3</a></i>;<br/>
<a NAME="E6:17"/>
( <font color="Maroon" title="c9">g1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c10">g2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="ref" href="waybel26.html#T2" target="_self" title="WAYBEL26:th.2">Th2</a></i>;<br/>
<a NAME="E7:17"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g1</font> &amp; <span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g2</font> )
 
<b>by </b><i><a class="ref" href="waybel26.html#D2" target="_self" title="WAYBEL26:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">fg1</font> = <font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g1</font>, <font color="Maroon" title="c12">fg2</font> = <font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g2</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">Z</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">Z</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> ) )</span><br/><b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">Z</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )</span><br/><b>then reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c13">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:17_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E5:17"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></i>;<br/><a NAME="E4:17_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font></span>)</span> &amp; <font color="Maroon" title="c9">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/><a NAME="E5:17_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/><b>hence </b><a NAME="E6:17_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">Z</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )
 ; <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="E10:17"/><b>then </b>
<font color="Maroon" title="c11">fg1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">fg2</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:17"/>
<span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E7:17"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="waybel26.html#T3" target="_self" title="WAYBEL26:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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