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

<b>let </b><font color="Maroon">LB</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>, <font color="Maroon">LB1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>;<br/>

<a NAME="E1:45_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#G1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10">Zero_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lopban_1.html#D15">LOPBAN_1:def 15</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:45_1_2"/><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">R</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">LB</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E3:45_1_2"/><i><font color="Green">E50</font></i>: 
 ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">R</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">LB1</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon">N</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E5:45_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">R</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">LB</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:45_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">LB</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ndiff_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">N1</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E9:45_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">N1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">R</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">LB1</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R1</font> being   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font><b> such that </b><br/><a NAME="E11:45_1_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">LB1</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">LB</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">L1</font> = <font color="Maroon">LB1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E14:45_1_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> holds  <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:45_1_2_1_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:45"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D6">LOPBAN_1:def 6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D6">LOPBAN_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:45"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:45_1_2_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:45_1_2_1_1_1_2_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> )
 ;<br/><a NAME="E2:45_1_2_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:45_1_2_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">N0</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E4:45_1_2_1_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">N0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">N0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:45_1_2_1_1_1_2"/><i><font color="Green">E67</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">N0</font> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D3">NFCONT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">n0</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="E8:45_1_2_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K6">/</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n0</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/><a NAME="E9:45_1_2_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E10:45_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:45_1_2_1_1_1_2"/>
<font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E12:45_1_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K6">/</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:45_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:45_1_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K6">/</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E14:45_1_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E15:45_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:45_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/><b>set </b><font color="Maroon">n1</font> = <font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E16:45_1_2_1_1_1_2"/>
 ex <font color="Olive">r</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> ) ) &amp; <font color="Olive">r</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E2:45_1_2_1_1_1_2_2"/><i><font color="Green">E71</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  <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>thus </b><a NAME="E3:45_1_2_1_1_1_2_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_1_2_2_1"><b>proof </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/>

<b>thus </b><a NAME="E1:45_1_2_1_1_1_2_2_1"/>
<font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a></i>;<br/>

<a NAME="E2:45_1_2_1_1_1_2_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:45_1_2_1_1_1_2_2_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E2:45_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:45_1_2_1_1_1_2_2_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E5:45_1_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green">E117</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:45_1_2_1_1_1_2_2_1"/>
<font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a></i>;<br/>

<a NAME="E7:45_1_2_1_1_1_2_2_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E8:45_1_2_1_1_1_2_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></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="E9:45_1_2_1_1_1_2_2_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E2:45_1_2"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="real_2.html#T201">REAL_2:201</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">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></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">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</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">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</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="rlvect_1.html#T28">RLVECT_1:28</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"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</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">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></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/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E5:45_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/>
<a NAME="E11:45_1_2_1_1_1_2_2_1"/><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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="txt" href="ndiff_1.html#E7:45_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:45_1_2_1_1_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E13:45_1_2_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:45_1_2_1_1_1_2_2"/>
<font color="Maroon">r</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:45_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="seq_4.html#T43">SEQ_4:43</a></i>;<br/>

<b>thus </b><a NAME="E5:45_1_2_1_1_1_2_2"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:45_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="seq_4.html#T44">SEQ_4:44</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">r</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E18:45_1_2_1_1_1_2"/><i><font color="Green">E120</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 <br/> ( <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> ) ) &amp; <font color="Maroon">r</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> = <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E20:45_1_2_1_1_1_2"/><i><font color="Green">E121</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  <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E21:45_1_2_1_1_1_2"/>
( <font color="Maroon">s</font> <a href="ndiff_1.html#NR1" target="_self">is_not_0</a>  &amp; <font color="Maroon">s</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_1_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_1_2_3_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/><b>assume </b><a NAME="E1:45_1_2_1_1_1_2_3_1"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:45_1_2_1_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E5:45_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E3:45_1_2_1_1_1_2_3_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T24">RLVECT_1:24</a></i>;<br/><b>hence </b><a NAME="E4:45_1_2_1_1_1_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:45"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1_2_1_1_1_2_3"/>
<font color="Maroon">s</font> <a href="ndiff_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_1_2_3_2"><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/><b>thus </b><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E5:45_1_2"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45_1_2_1_1_1_2_3"/>
( <font color="Maroon">s</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:45_1_2"><i><font color="Green">E48</font></i></a>, , </i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as  <a href="ndiff_1.html#V2" target="_self">convergent_to_0</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><div><i><font color="Green">E122</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_1_2_4"><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/><b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font>;<br/><i><font color="Green">E123</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E2:45_1_2_1_1_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E3:45_1_2_1_1_1_2_4"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="ndiff_1.html#E7:45_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E4:45_1_2_1_1_1_2_4"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E5:45_1_2_1_1_1_2_4"/><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E9:45_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><i><font color="Green">E127</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D6">LOPBAN_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
1 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:45_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D6">LOPBAN_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
1 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:45_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><a NAME="E8:45_1_2_1_1_1_2_4"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ndiff_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><a NAME="E9:45_1_2_1_1_1_2_4"/>
<font color="Maroon">R</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i/>;<br/><a NAME="E10:45_1_2_1_1_1_2_4"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/><a NAME="E11:45_1_2_1_1_1_2_4"/><b>then </b><i><font color="Green">E129</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E12:45_1_2_1_1_1_2_4"/>
<font color="Maroon">R1</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i/>;<br/><a NAME="E13:45_1_2_1_1_1_2_4"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/><a NAME="E14:45_1_2_1_1_1_2_4"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R1</font>
 ;<br/><a NAME="E15:45_1_2_1_1_1_2_4"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:45_1_2"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E16:45_1_2_1_1_1_2_4"/>
<font color="Maroon">s</font> <a href="ndiff_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i/>;<br/><a NAME="E17:45_1_2_1_1_1_2_4"/><b>then </b>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E18:45_1_2_1_1_1_2_4"/><b>then </b><i><font color="Green">E152</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">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><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> = 
<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">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E5:45_1_2"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</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"><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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:45_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span>

;<br/><b>then </b><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></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"><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="real_1.html#K2">"</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </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"><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="real_1.html#K4">*</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"><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="real_1.html#K2">"</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </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"><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="real_1.html#K6">/</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<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></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:45_1_2_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> 

;<br/><a NAME="E22:45_1_2_1_1_1_2_4"/><b>then </b><i><font color="Green">E153</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> 
 <b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/><i><font color="Green">E154</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <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>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E5:45_1_2"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <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>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <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>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E5:45_1_2"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <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>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E14:45_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E25:45_1_2_1_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="ndiff_1.html#E2:45_1_2_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E26:45_1_2_1_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><a NAME="E27:45_1_2_1_1_1_2_4"/><b>then </b>
<font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>;<br/><a NAME="E28:45_1_2_1_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E29:45_1_2_1_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E30:45_1_2_1_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E31:45_1_2_1_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/><a NAME="E32:45_1_2_1_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/><a NAME="E33:45_1_2_1_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T48">RLVECT_1:48</a></i>;<br/><a NAME="E34:45_1_2_1_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D6">NORMSP_1:def 6</a></i>;<br/><b>hence </b><a NAME="E35:45_1_2_1_1_1_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="normsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D8">NORMSP_1:def 8</a></i>;<br/></div><b>end;</b></div><a NAME="E24:45_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E155</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="normsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i><a class="ref" href="lopban_3.html#D5">LOPBAN_3:def 5</a></i>;<br/><a NAME="E25:45_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="normsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E26:45_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="normsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E7:45_1_2"><i><font color="Green">E53</font></i></a>, </i>;<br/><a NAME="E27:45_1_2_1_1_1_2"/><i><font color="Green">E157</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/><a NAME="E28:45_1_2_1_1_1_2"/><i><font color="Green">E158</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/><a NAME="E29:45_1_2_1_1_1_2"/><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E11:45_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ndiff_1.html#T2" target="_self">Th2</a>, <a class="ref" href="normsp_1.html#T35">NORMSP_1:35</a></i>;<br/><i><font color="Green">E160</font></i>:  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:45_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ndiff_1.html#T2" target="_self">Th2</a>, <a class="ref" href="normsp_1.html#T43">NORMSP_1:43</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/><i><font color="Green">E169</font></i>:  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="normsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><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#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="normsp_1.html#T45">NORMSP_1:45</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>
<br/>.= 
<font color="Maroon">L1</font> <a href="lopban_1.html#K9">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E9:45_1_2"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1_2_1"/>
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">LB</font> = 
 <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>
<br/>.= 
 <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
<br/>.= 
<font color="Maroon">LB1</font>
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>
;<br/>

<b>thus </b><a NAME="E16:45_1_2"/>
verum
 ;<br/>


</div>
