<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">seq1</font> is <a href="bhsp_3.html#V2" target="_self">bounded</a>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">seq2</font> is <a href="bhsp_3.html#V2" target="_self">bounded</a>
 ;<br/>

<b>consider </b><font color="Maroon">M1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:23"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">M1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">M2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:23"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">M2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E8:23"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M2</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:23"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">M1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">M2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:23_1"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 <b>by </b><i/>;<br/><a NAME="E2:23_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M2</font>
 <b>by </b><i/>;<br/><a NAME="E3:23_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">M2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:23_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="bhsp_1.html#K8">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>;<br/><a NAME="E5:23_1"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="bhsp_1.html#K8">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
 <b>by </b><i><a class="ref" href="bhsp_1.html#T36">BHSP_1:36</a></i>;<br/><b>hence </b><a NAME="E6:23_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="bhsp_1.html#K8">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">M2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:23"/>
<font color="Maroon">seq1</font> <a href="bhsp_1.html#K8">+</a> <font color="Maroon">seq2</font> is <a href="bhsp_3.html#V2" target="_self">bounded</a>
 <b>by </b><i>, </i>;<br/>


</div>
