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

<b>let </b><font color="Maroon">X</font> be   <a href="clopban1.html#NM2">ComplexBanachSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> is    <a href="ncfcont2.html#M1" target="_self">contraction</a> of <font color="Maroon">X</font>
 ;<br/>

<b>consider </b><font color="Maroon">x0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:51"/><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp; ( 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">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S3">RECDEF_1:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon">K</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:51"/><i><font color="Green">E52</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E7:51"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>and </b><br/><a NAME="E8:51"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E9:51"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E10:51"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:51_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:51_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:51"/><i><font color="Green">E59</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<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E12:51"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E14:51"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<i><font color="Green">E62</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> 1

<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/>
<a NAME="E2:51_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E3:51_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:51_2_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:51_2_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:51_2_2"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:51_2_2"/>
<font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:51_2_2"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ncfcont2.html#D4" target="_self">Def4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:51_2_2"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<b>hence </b><a NAME="E8:51_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="ncfcont2.html#D2" target="_self">Def2</a>, <a class="ref" href="ncfcont2.html#D3" target="_self">Def3</a>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:51_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  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="ncfcont2.html#D1" target="_self">Def1</a>, );<br/></i>
<b>hence </b><a NAME="E5:51_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  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Olive">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:51"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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/><i><font color="Green">E89</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="clvect_1.html#T103">CLVECT_1:103</a></i>
;<br/><b>thus </b><a NAME="E2:51_3_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:51_3"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:51_3_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:51_3_2_1"/><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E2:51_3_2_1"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T112">CLVECT_1:112</a></i>;<br/><a NAME="E3:51_3_2_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:51_3_2_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D2" target="_self">Def2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:51_3_2_1"/>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span> <a href="real_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="clvect_1.html#K5">||.</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</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"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</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"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/><b>hence </b><a NAME="E7:51_3_2_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D3" target="_self">Def3</a>, <a class="ref" href="ncfcont2.html#D4" target="_self">Def4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:51_3_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:51_3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="ncfcont2.html#D1" target="_self">Def1</a>, );<br/></i>
<b>hence </b><a NAME="E5:51_3"/>
for <font color="Olive">k</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:51"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:51_4_1"/><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E2:51_4_1"/><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:51_4_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E2:51_4_1_1"/><i><font color="Green">E103</font></i>: 
1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E3:51_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="ncfcont2.html#D2" target="_self">Def2</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>hence </b><a NAME="E4:51_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D3" target="_self">Def3</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:51_4_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T106">CLVECT_1:106</a></i>;<br/><a NAME="E4:51_4_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_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">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E5:51_4_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_4"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:51"/>
<font color="Maroon">g</font> is <a href="csspace3.html#V1">CCauchy</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:51_5_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:51_5_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:51_5_1"/><i><font color="Green">E107</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , , </i>;<br/><a NAME="E5:51_5_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E6:51_5_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="ncfcont2.html#D2" target="_self">Def2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:51_5_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T76">XCMPLX_1:76</a></i>;<br/><b>take </b><font color="Maroon">nn</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:51_5_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">nn</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E2:51_5_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">nn</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E3:51_5_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D4" target="_self">Def4</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:51_5_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D5" target="_self">Def5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>consider </b><font color="Maroon">k1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:51_5_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D6" target="_self">Def6</a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">k2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:51_5_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="ncfcont2.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k2</font> = <font color="Maroon">k2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E11:51_5_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E12:51_5_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="ncfcont2.html#D3" target="_self">Def3</a>, <a class="txt" href="ncfcont2.html#E1:51"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:51_5_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E14:51_5_1_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="ncfcont2.html#D3" target="_self">Def3</a>, <a class="txt" href="ncfcont2.html#E4:51"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E15:51_5_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="ncfcont2.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:51_5_1"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">nn</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">nn</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_5"/>
<font color="Maroon">g</font> is <a href="csspace3.html#V1">CCauchy</a>
 <b>by </b><i><a class="ref" href="csspace3.html#T10">CSSPACE3:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:51"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">g</font> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban1.html#D14">CLOPBAN1:def 14</a></i>;<br/>
<a NAME="E19:51"/><i><font color="Green">E117</font></i>: 
 <a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_6"><b>proof </b></a><div class="add">

<a NAME="E1:51_6"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">g</font> <a href="clopban3.html#K3">^\</a> 1 is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="clopban3.html#K3">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="clopban3.html#T13">CLOPBAN3:13</a></i>;<br/>
<div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_6_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/><span class="p1"><a href="clvect_1.html#K5">||.</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">g</font> <a href="clopban3.html#K3">^\</a> 1</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="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clopban3.html#D6">CLOPBAN3:def 6</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E2:51_6_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</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">g</font> <a href="clopban3.html#K3">^\</a> 1</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="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:51_6"/><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="clvect_1.html#K8">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="clvect_1.html#T120">CLVECT_1:120</a></i>;<br/>
<a NAME="E4:51_6"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">K</font> <a href="seq_1.html#K14">(#)</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="clvect_1.html#K8">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="ncfcont2.html#D4" target="_self">Def4</a>, <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>
<i><font color="Green">E140</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="seq_1.html#K14">(#)</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="clvect_1.html#K8">-</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> = 
<font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="clvect_1.html#K8">-</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont2.html#D4" target="_self">Def4</a>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
<br/>.= 
<font color="Maroon">K</font> <a href="real_1.html#K4">*</a> 0
<b>by </b><i>, <a class="ref" href="clvect_1.html#T120">CLVECT_1:120</a></i>
<br/>.= 
0

;<br/>
<a NAME="E6:51_6"/>
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</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">g</font> <a href="clopban3.html#K3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:51_6_3"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:51_6_3"/><i><font color="Green">E142</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="seq_1.html#K14">(#)</a> <span class="p4"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="clvect_1.html#K8">-</a> <span class="p0">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D5" target="_self">Def5</a>, <a class="ref" href="ncfcont2.html#D6" target="_self">Def6</a>, <a class="ref" href="ncfcont2.html#T1" target="_self">Th1</a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_6_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:51_6_3_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:51_6_3_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">K</font> <a href="seq_1.html#K14">(#)</a> <span class="p4"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="clvect_1.html#K8">-</a> <span class="p0">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="ncfcont2.html#E1:51"><i><font color="Green">E46</font></i></a>, <a class="txt" href="ncfcont2.html#E4:51"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E3:51_6_3_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="clvect_1.html#K8">-</a> <span class="p5">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/><a NAME="E4:51_6_3_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="clvect_1.html#K8">-</a> <span class="p5">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/><a NAME="E5:51_6_3_1"/><b>then </b><i><font color="Green">E144</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D13">CLVECT_1:def 13</a></i>;<br/><a NAME="E6:51_6_3_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E7:51_6_3_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="ncfcont2.html#T2" target="_self">Th2</a>, <a class="ref" href="ncfcont2.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:51_6_3_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</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">g</font> <a href="clopban3.html#K3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E9:51_6_3_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</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">g</font> <a href="clopban3.html#K3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:51_6_3"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</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">g</font> <a href="clopban3.html#K3">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:51_6"/>
 <a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D2" target="_self">Def2</a>, <a class="ref" href="clvect_1.html#D17">CLVECT_1:def 17</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon">xp</font> =  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">g</font>;<br/>

<div><i><font color="Green">E147</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_7"><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">X</font>;<br/><b>assume </b><a NAME="E1:51_7"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:51_7"/><i><font color="Green">E149</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Olive">k</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_7_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<i><font color="Green">E150</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> 1

<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/>
<a NAME="E2:51_7_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="ncfcont2.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:51_7_1"/><i><font color="Green">E152</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_7_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_7_1_2"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:51_7_1_2"/><i><font color="Green">E154</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:51_7_1_2"/>
<font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="ncfcont2.html#E1:51"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E4:51_7_1_2"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ncfcont2.html#E4:51"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:51_7_1_2"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<b>hence </b><a NAME="E6:51_7_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , <a class="ref" href="ncfcont2.html#D3" target="_self">Def3</a>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:51_7_1"/>
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<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="ncfcont2.html#D6" target="_self">Def6</a>, <a class="ref" href="ncfcont2.html#T1" target="_self">Th1</a>);<br/></i>
<b>hence </b><a NAME="E5:51_7_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Olive">k</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:51_7"/><i><font color="Green">E155</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:51_7_2"/><i><font color="Green">E156</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:51_7_2"/><i><font color="Green">E157</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="ncfcont2.html#D6" target="_self">Def6</a>, </i>;<br/>
<a NAME="E4:51_7_2"/><i><font color="Green">E158</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E5:51_7_2"/><i><font color="Green">E159</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="ncfcont2.html#T1" target="_self">Th1</a>, <a class="txt" href="ncfcont2.html#E1:51"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:51_7_2"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ncfcont2.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:51_7_2"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">xp</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="ncfcont2.html#E4:51"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E8:51_7_2"/>
verum
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:51_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xp</font>
 <b>by </b><i><a class="ref" href="ncfcont2.html#D5" target="_self">Def5</a>, </i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E21:51"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xp</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> holds <br/><font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, <a class="ref" href="ncfcont2.html#D2" target="_self">Def2</a></i>;<br/>


</div>
