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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">b</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:108"/><span class="lab"><font color="Green" title="E88">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:108"/><span class="lab"><font color="Green" title="E89">A2</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">b</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">p</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a><span class="kw"> such that </span><br/><a NAME="E4:108"/><span class="lab"><font color="Green" title="E90">A3</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">and </span><br/><a NAME="E5:108"/><span class="lab"><font color="Green" title="E91">A4</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="rat_1.html#T7" title="RAT_1:th.7">RAT_1:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">q</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a><span class="kw"> such that </span><br/><a NAME="E7:108"/><span class="lab"><font color="Green" title="E92">A5</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">and </span><br/><a NAME="E8:108"/><span class="lab"><font color="Green" title="E93">A6</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="ref" href="rat_1.html#T7" title="RAT_1:th.7">RAT_1:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">s2</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><span class="kw"> such that </span><br/><a NAME="E10:108"/><span class="lab"><font color="Green" title="E94">A7</font></span>: 
<font color="Maroon" title="c6">s2</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E11:108"/><span class="lab"><font color="Green" title="E95">A8</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s2</font>
 <span class="kw">and </span><br/><a NAME="E12:108"/><span class="lab"><font color="Green" title="E96">A9</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">s2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T67" target="_self" title="PREPOWER:th.67">Th79</a></span>;<br/>
<a NAME="E13:108"/><span class="lab"><font color="Green" title="E97">A10</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E8:108"><span class="lab"><font color="Green" title="E93">A6</font></span></a>, <a class="ref" href="prepower.html#T64" target="_self" title="PREPOWER:th.64">Th75</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font></span><br/><a NAME="E1:108_1"/>
<font color="Maroon" title="c6">s2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:108"><span class="lab"><font color="Green" title="E96">A9</font></span></a></span>;<br/><a NAME="E2:108_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">s2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108"><span class="lab"><font color="Green" title="E92">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:108_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="ref" href="prepower.html#T63" target="_self" title="PREPOWER:th.63">Th74</a></span>;<br/><span class="kw">hence </span><a NAME="E4:108_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D6" target="_self" title="PREPOWER:def.6">Def7</a></span>; <a class="txt" 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="E15:108"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A11</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E10:108"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="ref" href="prepower.html#T2" target="_self" title="PREPOWER:th.2">Th3</a>, <a class="ref" href="prepower.html#T69" target="_self" title="PREPOWER:th.69">Th82</a></span>;<br/>
<a NAME="E16:108"/>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c6">s2</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E10:108"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="ref" href="prepower.html#T69" target="_self" title="PREPOWER:th.69">Th82</a></span>;<br/>
<a NAME="E17:108"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E10:108"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="txt" href="#E11:108"><span class="lab"><font color="Green" title="E95">A8</font></span></a>, <a class="txt" href="#E15:108"><span class="lab"><font color="Green" title="E98">A11</font></span></a>, <a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def8</a></span>;<br/>
<a NAME="E18:108"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A12</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:108"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">s1</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><span class="kw"> such that </span><br/><a NAME="E20:108"/><span class="lab"><font color="Green" title="E100">A13</font></span>: 
<font color="Maroon" title="c7">s1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E21:108"/><span class="lab"><font color="Green" title="E101">A14</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c7">s1</font>
 <span class="kw">and </span><br/><a NAME="E22:108"/><span class="lab"><font color="Green" title="E102">A15</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T68" target="_self" title="PREPOWER:th.68">Th80</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c7">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font></span><br/><a NAME="E1:108_2"/>
<font color="Maroon" title="c7">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:108"><span class="lab"><font color="Green" title="E102">A15</font></span></a></span>;<br/><a NAME="E2:108_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:108_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="ref" href="prepower.html#T63" target="_self" title="PREPOWER:th.63">Th74</a></span>;<br/><span class="kw">hence </span><a NAME="E4:108_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c7">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D6" target="_self" title="PREPOWER:def.6">Def7</a></span>; <a class="txt" 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="E24:108"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A16</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c7">s1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E20:108"><span class="lab"><font color="Green" title="E100">A13</font></span></a>, <a class="ref" href="prepower.html#T1" target="_self" title="PREPOWER:th.1">Th2</a>, <a class="ref" href="prepower.html#T69" target="_self" title="PREPOWER:th.69">Th82</a></span>;<br/>
<a NAME="E25:108"/>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c7">s1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E20:108"><span class="lab"><font color="Green" title="E100">A13</font></span></a>, <a class="ref" href="prepower.html#T69" target="_self" title="PREPOWER:th.69">Th82</a></span>;<br/>
<a NAME="E26:108"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E20:108"><span class="lab"><font color="Green" title="E100">A13</font></span></a>, <a class="txt" href="#E21:108"><span class="lab"><font color="Green" title="E101">A14</font></span></a>, <a class="txt" href="#E24:108"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:108"/>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:108"><span class="lab"><font color="Green" title="E99">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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