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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>;<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>;<br/><b>let </b><font color="Maroon" title="c4">k</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> ;<br/>







<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E23">A1</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>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">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#D2" target="_self" title="BHSP_3:def.2">BHSP_3:def 2</a><br/>

<b>assume </b><a NAME="E2:56"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>then consider </b><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> <b> such that </b><br/><a NAME="E4:56"/><i><font color="Green" title="E24">A2</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">m1</font> holds <br/> <a href="bhsp_1.html#K5" title="BHSP_1:func.5">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="c5">r</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E1:56"><i><font color="Green" title="E23">A1</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">m</font> = <font color="Maroon" title="c6">m1</font>;<br/>

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E5:56"/><i><font color="Green" title="E25">A3</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">m</font>
 ;<br/>

<a NAME="E6:56"/>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E7:56"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font>
 
<b>by </b><i><a class="txt" href="bhsp_3.html#E5:56"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:56"/><b>then </b>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="bhsp_3.html#E4:56"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<a NAME="E9:56"/><b>then </b>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="ref" href="bhsp_3.html#D5" target="_self" title="BHSP_3:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E10:56"/>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <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="c5">r</font>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D5" target="_self" title="BHSP_3:def.5">Def5</a></i>;<br/>


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