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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">C</font> is <a href="yellow18.html#V2" title="YELLOW18:attr.2">para-functional</a> iff for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b1">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b2">a2</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E1:69"/>
( <font color="Maroon" title="c1">C</font> is <a href="yellow18.html#V2" title="YELLOW18:attr.2">para-functional</a> implies for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b1">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b2">a2</font></span>)</span> )
  <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">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b1">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b2">a2</font></span>)</span> ) implies <font color="Maroon" title="c1">C</font> is <a href="yellow18.html#V2" title="YELLOW18:attr.2">para-functional</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c2">F</font> being   <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">C</font><b> such that </b><a NAME="E2:69_1"/><i><font color="Green" title="E33">A1</font></i>: 
for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a2</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D7" target="_self" title="YELLOW18:def.7">YELLOW18:def 7</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">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b1">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b2">a2</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c4">a2</font></span>)</span></span><br/>

<a NAME="E3:69_1"/><i><font color="Green" title="E34">A2</font></i>: 
(  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c3">a1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c3">a1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a1</font></span>)</span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a2</font></span>)</span> &amp;  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:69_1"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:69_1"/><i><font color="Green" title="E35">A3</font></i>: 
(  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a1</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:69_1"/><i><font color="Green" title="E36">A4</font></i>: 
(  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a2</font> )
 <b>by </b><i><a class="txt" href="yellow18.html#E3:69_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E8:69_1"/>
(  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a2</font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E5:69_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="yellow18.html#E7:69_1"><i><font color="Green" title="E36">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:69_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c4">a2</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E2:69_1"><i><font color="Green" title="E33">A1</font></i></a></i>; <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>

<b>assume </b><a NAME="E2:69"/><i><font color="Green" title="E33">A5</font></i>: 
for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b1">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b2">a2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">C</font> is <a href="yellow18.html#V2" title="YELLOW18:attr.2">para-functional</a></span><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c1">C</font> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c2">F</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:69"/><i><font color="Green" title="E34">A6</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c2">F</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D7" target="_self" title="YELLOW18:def.7">YELLOW18:def 7</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">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a2</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font></span>)</span></span><br/>

<a NAME="E5:69"/>
( <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E4:69"><i><font color="Green" title="E34">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:69"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E2:69"><i><font color="Green" title="E33">A5</font></i></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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