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<b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:18"/>
<font color="Maroon">seq1</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">seq</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Ns</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">seq1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns</font>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D4">BHSP_3:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:18"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:18"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:18"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lopban_3.html#T7">LOPBAN_3:7</a></i>;<br/>
<b>hence </b><a NAME="E8:18"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 <b>by </b><i/>;<br/>


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