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

<b>let </b><font color="Maroon">X</font> be   <a href="lopban_2.html#NM3">Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">s'</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="E1:11"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">s</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">s'</font> is <a href="normsp_1.html#V4">convergent</a>
 ;<br/>

<a NAME="E3:11"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">s</font> <a href="lopban_3.html#K8">*</a> <font color="Maroon">s'</font> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">g1</font> =  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">g2</font> =  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span>;<br/>
<a NAME="E4:11"/>
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="normsp_1.html#T39">NORMSP_1:39</a></i>;<br/>
<a NAME="E5:11"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T27">SEQ_2:27</a></i>;<br/>
<a NAME="E6:11"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">R</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:11"/><i><font color="Green">E50</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="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>
<a NAME="E9:11"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> 1
 
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/>
<a NAME="E10:11"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> 1
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E11:11"/><b>then </b><i><font color="Green">E51</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_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="E1:11_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/><b>hence </b><a NAME="E2:11_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E14:11"/><i><font color="Green">E54</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E15:11"/><b>then </b><i><font color="Green">E55</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font>
 
<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="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:11_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:11_2"/><b>then </b><i><font color="Green">E64</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>then consider </b><font color="Maroon">n1</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="E4:11_2"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/><b>consider </b><font color="Maroon">n2</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="E6:11_2"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E7:11_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E8:11_2"/><i><font color="Green">E68</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/><a NAME="E9:11_2"/><i><font color="Green">E69</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/><a NAME="E10:11_2"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E11:11_2"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:11_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:11_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E14:11_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:11_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i/>;<br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="lopban_3.html#K8">*</a> <font color="Maroon">s'</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_3.html#D11">LOPBAN_3:def 11</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
;<br/><a NAME="E17:11_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="lopban_3.html#K8">*</a> <font color="Maroon">s'</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><a NAME="E18:11_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E19:11_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">R</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T98">XREAL_1:98</a></i>;<br/><a NAME="E20:11_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>;<br/><a NAME="E21:11_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">R</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E22:11_2"/><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>;<br/><a NAME="E23:11_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E24:11_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p5">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <font color="Maroon">R</font></span>)</span>

<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/><a NAME="E26:11_2"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E27:11_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="lopban_3.html#K8">*</a> <font color="Maroon">s'</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</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="E17:11"/>
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="lopban_3.html#K8">*</a> <font color="Maroon">s'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/>


</div>
