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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">seq</font>, <font color="Maroon" title="c3">seq1</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>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="bhsp_3.html#V2" title="BHSP_3:attr.2">bounded</a>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c3">seq1</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">seq</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">M1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:38"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c4">M1</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> 
 <b>and </b><br/><a NAME="E5:38"/><i><font color="Green" title="E19">A4</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>  holds  <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E1:38"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="bhsp_3.html#D3" target="_self" title="BHSP_3:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">Nseq</font> being  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E7:38"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c3">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K1" title="BHSP_3:func.1">*</a> <font color="Maroon" title="c5">Nseq</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E2:38"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="bhsp_3.html#D4" target="_self" title="BHSP_3:def.4">Def4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c4">M1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bhsp_3.html#D3" target="_self" title="BHSP_3:def.3">BHSP_3:def 3</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><a NAME="E1:38_1"/>
<font color="Maroon" title="c3">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Nseq</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E7:38"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="bhsp_3.html#T26" target="_self" title="BHSP_3:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E2:38_1"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E5:38"><i><font color="Green" title="E19">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:38"/>
( <font color="Maroon" title="c6">M</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>  &amp; ( 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  <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> ) )
 <b>by </b><i><a class="txt" href="bhsp_3.html#E4:38"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>


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