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<div class="add">

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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"> for <font color="Olive" title="b1">W</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">W</font><br/> for <font color="Olive" title="b4">A'</font>, <font color="Olive" title="b5">B'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">A'</font> &amp; <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">B'</font> holds <br/> <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Olive" title="b4">A'</font>,<font color="Olive" title="b5">B'</font></span><br/><b>let </b><font color="Maroon" title="c2">W</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>; <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">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">W</font><br/> for <font color="Olive" title="b3">A'</font>, <font color="Olive" title="b4">B'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A'</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B'</font> holds <br/> <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Olive" title="b3">A'</font>,<font color="Olive" title="b4">B'</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">W</font>; <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">A'</font>, <font color="Olive" title="b2">B'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A'</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B'</font> holds <br/> <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">B'</font></span><br/><b>let </b><font color="Maroon" title="c5">A'</font>, <font color="Maroon" title="c6">B'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A'</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B'</font> implies  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font> )</span><br/>







<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A'</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font> ) &amp; ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> ) )</span><br/><b>thus </b><a NAME="E1:34_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font></span><br/>

<a NAME="E2:34_1_1"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T17" target="_self" title="ENS_1:th.17">Th17</a></i>;<br/>
<a NAME="E3:34_1_1"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:34_1_1"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c2">W</font> <b>by </b><i><a class="txt" href="ens_1.html#E1:34_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="ens_1.html#E2:34_1_1"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/>
<a NAME="E5:34_1_1"/>
 <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T7" target="_self" title="ENS_1:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">m'</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="ens_1.html#E3:34_1_1"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<a NAME="E7:34_1_1"/>
( <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c8">m</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c8">m</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c9">m'</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c9">m'</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:34_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="ens_1.html#T8" target="_self" title="ENS_1:th.8">Th8</a>, <a class="ref" href="ens_1.html#T16" target="_self" title="ENS_1:th.16">Th16</a></i>;<br/>
<a NAME="E8:34_1_1"/><b>then </b>
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c9">m'</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c9">m'</font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E10"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:34_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="ens_1.html#T19" target="_self" title="ENS_1:th.19">Th19</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="E2:34_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><br/><a NAME="E3:34_1"/>
 <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="ens_1.html#T17" target="_self" title="ENS_1:th.17">Th17</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">m'</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="ens_1.html#E2:34_1"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/><a NAME="E5:34_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c8">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">m'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="ens_1.html#E2:34_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="ens_1.html#T16" target="_self" title="ENS_1:th.16">Th16</a></i>;<br/><a NAME="E6:34_1"/><b>then </b>
( ( <font color="Maroon" title="c6">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c5">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c8">m'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E2:34_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="ens_1.html#E29"><i><font color="Green" title="E20">Lm4</font></i></a></i>;<br/><b>hence </b><a NAME="E7:34_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="ens_1.html#E5:34_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="ens_1.html#T15" target="_self" title="ENS_1:th.15">Th15</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>hence </b><a NAME="E3:34"/>
 <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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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