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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font>, <font color="Olive" title="b3">seq3</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">seq1</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Olive" title="b2">seq2</font> &amp; <font color="Olive" title="b2">seq2</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Olive" title="b3">seq3</font> holds <br/><font color="Olive" title="b1">seq1</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Olive" title="b3">seq3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font>, <font color="Maroon" title="c4">seq3</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</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="c2">seq1</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Maroon" title="c3">seq2</font> &amp; <font color="Maroon" title="c3">seq2</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Maroon" title="c4">seq3</font> implies <font color="Maroon" title="c2">seq1</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Maroon" title="c4">seq3</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c2">seq1</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Maroon" title="c3">seq2</font>
 <span class="kw">and </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c3">seq2</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Maroon" title="c4">seq3</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="c2">seq1</font> <a href="bhsp_3.html#R2" title="BHSP_3:pred.2">is_compared_to</a> <font color="Maroon" title="c4">seq3</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="bhsp_3.html#D2" target="_self" title="BHSP_3:def.2">BHSP_3:def 2</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="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">m</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>  st <br/> for <font color="Olive" title="b2">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>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:15"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</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>  st <br/> for <font color="Olive" title="b2">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>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E4:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T215" title="XREAL_1:th.215">XREAL_1:215</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">m1</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> <span class="kw"> such that </span><br/><a NAME="E6:15"/><span class="lab"><font color="Green" title="E10">A4</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">m1</font> holds <br/> <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="bhsp_3.html#D2" target="_self" title="BHSP_3:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">m2</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> <span class="kw"> such that </span><br/><a NAME="E8:15"/><span class="lab"><font color="Green" title="E11">A5</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m2</font> holds <br/> <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="bhsp_3.html#D2" target="_self" title="BHSP_3:def.2">Def2</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c6">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">m2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">m</font> holds <br/> <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">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"> ( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">m</font> implies  <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E9:15"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E10:15"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E11:15"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A7</font></span>: 
 <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>;<br/>
<a NAME="E13:15"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
 <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#T35" title="BHSP_1:th.35">BHSP_1:35</a></span>;<br/>
<a NAME="E14:15"/>
<font color="Maroon" title="c6">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">m2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E15:15"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:15"/><span class="kw">then </span>
 <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>;<br/>
<a NAME="E17:15"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:15"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:15"/>
 <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E14">A8</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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