<?xml version="1.0"?>
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<span class="kw">set </span><font color="Maroon" title="c5">lf</font> =  <a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">thus </span><a NAME="E1:109_1_1"/>
 <a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font> is  <a href="bilinear.html#V1" title="BILINEAR:attr.1">additiveFAF</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font> is  <a href="bilinear.html#V3" title="BILINEAR:attr.3">homogeneousFAF</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="bilinear.html#D12" target="_self" title="BILINEAR:def.12">BILINEAR:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">flf</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E2:109_1_1_1"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">w</font>, <font color="Maroon" title="c10">t</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">w</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">t</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">w</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">w</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c10">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T28" target="_self" title="BILINEAR:th.28">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c10">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c10">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c10">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="bilinear.html#D14" target="_self" title="BILINEAR:def.14">BILINEAR:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font> is  <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">homogeneous</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">flf</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:109_1_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">w</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="hahnban1.html#D12" title="HAHNBAN1:def.12">HAHNBAN1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c10">r</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Scalar</a> of ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font></span>)</span></span><br/>



<span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></span>
<br/>.= 
<font color="Maroon" title="c10">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T33" target="_self" title="BILINEAR:th.33">Th33</a></span>
<br/>.= 
<font color="Maroon" title="c10">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="bilinear.html#D21" target="_self" title="BILINEAR:def.21">Def21</a></span>
<br/>.= 
<font color="Maroon" title="c10">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T9" target="_self" title="BILINEAR:th.9">Th9</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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