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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</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> is <a href="bhsp_3.html#V1" title="BHSP_3:attr.1">Cauchy</a> &amp; <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> holds <br/><font color="Olive" title="b2">seq2</font> is <a href="bhsp_3.html#V1" title="BHSP_3:attr.1">Cauchy</a></span><br/><b>let </b><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">seq1</font> is <a href="bhsp_3.html#V1" title="BHSP_3:attr.1">Cauchy</a> &amp; <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> implies <font color="Maroon" title="c3">seq2</font> is <a href="bhsp_3.html#V1" title="BHSP_3:attr.1">Cauchy</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="bhsp_3.html#V1" title="BHSP_3:attr.1">Cauchy</a>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green" title="E8">A2</font></i>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">seq2</font> is <a href="bhsp_3.html#V1" title="BHSP_3:attr.1">Cauchy</a></span><br/>

<b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="bhsp_3.html#D1" target="_self" title="BHSP_3:def.1">BHSP_3:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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">k</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>, <font color="Olive" title="b3">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 <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">k</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</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#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> )</span><br/>

<b>assume </b><a NAME="E3:18"/>
<font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k</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>, <font color="Olive" title="b3">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 <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">k</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</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#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/>

<a NAME="E4:18"/><b>then </b><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T224" title="XREAL_1:th.224">XREAL_1:224</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">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> <b> such that </b><br/><a NAME="E6:18"/><i><font color="Green" title="E10">A4</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">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 <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">m1</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">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#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 <b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="bhsp_3.html#D1" target="_self" title="BHSP_3:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E8:18"/><i><font color="Green" title="E11">A5</font></i>: 
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">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="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</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#K2" title="NORMSP_1:func.2">.</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="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 <b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E4:18"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="bhsp_3.html#D2" target="_self" title="BHSP_3:def.2">Def2</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c5">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m2</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">n</font>, <font color="Olive" title="b2">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 <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">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</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#K2" title="NORMSP_1:func.2">.</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#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/>

<b>let </b><font color="Maroon" title="c8">n</font>, <font color="Maroon" title="c9">m</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</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="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> )</span><br/>



<b>assume </b><a NAME="E9:18"/><i><font color="Green" title="E12">A6</font></i>: 
( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/>

<a NAME="E10:18"/>
<font color="Maroon" title="c5">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">m1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E11:18"/><b>then </b>
( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">m1</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">m1</font> )
 
<b>by </b><i><a class="txt" href="#E9:18"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:18"/><b>then </b><i><font color="Green" title="E13">A7</font></i>: 
 <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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 
<b>by </b><i><a class="txt" href="#E6:18"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>
<a NAME="E13:18"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">m2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E14:18"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">m2</font>
 
<b>by </b><i><a class="txt" href="#E9:18"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:18"/><b>then </b><i><font color="Green" title="E15">A9</font></i>: 
 <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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 
<b>by </b><i><a class="txt" href="#E8:18"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<a NAME="E16:18"/>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">m2</font>
 
<b>by </b><i><a class="txt" href="#E9:18"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="#E13:18"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:18"/><b>then </b><i><font color="Green" title="E16">A10</font></i>: 
 <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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 
<b>by </b><i><a class="txt" href="#E8:18"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<a NAME="E18:18"/><i><font color="Green" title="E17">A11</font></i>: 
<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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">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="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</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="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span>
 
<b>by </b><i><a class="txt" href="#E12:18"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="#E15:18"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E19:18"/>
 <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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</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="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">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="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T42" title="BHSP_1:th.42">BHSP_1:42</a></i>;<br/>
<a NAME="E20:18"/><b>then </b>
 <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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></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="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span>
 
<b>by </b><i><a class="txt" href="#E18:18"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E21:18"/><b>then </b><i><font color="Green" title="E18">A12</font></i>: 
<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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</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="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span>
 
<b>by </b><i><a class="txt" href="#E17:18"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E22:18"/>
 <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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</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="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</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="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T42" title="BHSP_1:th.42">BHSP_1:42</a></i>;<br/>
<b>hence </b><a NAME="E23:18"/>
 <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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="#E21:18"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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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