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

<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="ali2.html#M1">contraction</a> of <font color="Maroon">M</font>;<br/>



<b>consider </b><font color="Maroon">L</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:14"/><i><font color="Green">E44</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> &amp; <font color="Maroon">L</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green">E45</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">M</font> holds   <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="ali2.html#D1">ALI2:def 1</a></i>;<br/>
<a NAME="E4:14"/><i><font color="Green">E48</font></i>: 
1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E5:14"/><i><font color="Green">E49</font></i>: 
1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E6:14"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">M</font> is <a href="tbsp_1.html#V4" target="_self">complete</a>
 ;<br/>

<a NAME="E7:14"/>
 ex <font color="Olive">S1</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">M</font> st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">S1</font> <a href="funct_2.html#K3">.</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">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>
<a NAME="E2:14_1"/>
 ex <font color="Olive">h</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">z</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="Olive">h</font> <a href="funct_2.html#K3">.</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="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>) ) )
 
<b>from </b><i><a class="ref" href="recdef_1.html#S4">RECDEF_1:sch 4</a>();<br/></i>
<b>then consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:14_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">z</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">h</font> <a href="funct_2.html#K3">.</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">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

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

<b>thus </b><a NAME="E5:14_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</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">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">S1</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E9:14"/><i><font color="Green">E60</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">S1</font> <a href="funct_2.html#K3">.</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">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>
<b>set </b><font color="Maroon">r</font> =  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>;<br/>
<a NAME="E10:14"/><i><font color="Green">E62</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1"/>
( 0 <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> or 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_1"/><i><font color="Green">E63</font></i>: 
0 <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 ;<br/><div class="add"><i><font color="Green">E64</font></i>: <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> = 
<font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0
<b>by </b><i>, <a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>
;<br/><a NAME="E3:14_2_1_1"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E1:14_2_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<b>assume </b><a NAME="E2:14_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0
 ;<br/>

<a NAME="E3:14_2_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>
<a NAME="E4:14_2_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E5:14_2_1_1_2"/><b>then </b>
<font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T145">REAL_2:145</a></i>;<br/>
<b>hence </b><a NAME="E6:14_2_1_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:14_2_1_1"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_2"/><i><font color="Green">E76</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 ;<br/><div class="add"><a NAME="E2:14_2_1_2"/><i><font color="Green">E77</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   <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</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="14_2_1_2_1"><b>proof </b></a><div class="add">

<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> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:14_2_1_2_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_1"><b>proof </b></a><div class="add">

 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1

<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</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/>
<b>hence </b><a NAME="E2:14_2_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:14_2_1_2_1"/><i><font color="Green">E79</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="14_2_1_2_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="E1:14_2_1_2_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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 href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E3:14_2_1_2_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:14_2_1_2_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</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/>
<font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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>, <a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/>
<b>hence </b><a NAME="E6:14_2_1_2_1_2"/>
<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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:14_2_1_2_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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div><a NAME="E3:14_2_1_2"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<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 class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_2"><b>proof </b></a><div class="add">

<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>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>;<br/>
<a NAME="E1:14_2_1_2_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_2_1"><b>proof </b></a><div class="add">

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

<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> = 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
<br/>.= 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>

;<br/>
<b>hence </b><a NAME="E2:14_2_1_2_2_1"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:14_2_1_2_2"/><i><font color="Green">E85</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_2_1_2_2_2"/><i><font color="Green">E86</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   <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
 ;<br/>

<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/>

<a NAME="E2:14_2_1_2_2_2"/><i><font color="Green">E87</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><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> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E4:14_2_1_2_2_2"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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 class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:14_2_1_2_2_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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 class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L</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> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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="E7:14_2_1_2_2_2"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">L</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> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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>thus </b><a NAME="E3:14_2_1_2_2"/>
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>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div><a NAME="E4:14_2_1_2"/><i><font color="Green">E90</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   <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Olive">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">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_3"><b>proof </b></a><div class="add">

<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> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">a<sub>1</sub></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">L</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:14_2_1_2_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_3_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> 0</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">L</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</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">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
<br/>.= 
0

;<br/>
<b>hence </b><a NAME="E2:14_2_1_2_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:14_2_1_2_3"/><i><font color="Green">E92</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="14_2_1_2_3_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="E1:14_2_1_2_3_2"/><i><font color="Green">E93</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:14_2_1_2_3_2"/><i><font color="Green">E94</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E3:14_2_1_2_3_2"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:14_2_1_2_3_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</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">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></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">L</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</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">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</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">1 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</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">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <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">L</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="E6:14_2_1_2_3_2"/>
<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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:14_2_1_2_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>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div><a NAME="E5:14_2_1_2"/><i><font color="Green">E96</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   <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_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/>

<a NAME="E1:14_2_1_2_4"/><i><font color="Green">E97</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:14_2_1_2_4"/>
( <font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:14_2_1_2_4"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/>
<a NAME="E4:14_2_1_2_4"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font></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">L</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E5:14_2_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</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">L</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</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="E6:14_2_1_2_4"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:14_2_1_2_4"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:14_2_1_2"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<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 class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</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="14_2_1_2_5"><b>proof </b></a><div class="add">

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

<a NAME="E1:14_2_1_2_5"/><i><font color="Green">E99</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:14_2_1_2_5"/><i><font color="Green">E100</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:14_2_1_2_5"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:14_2_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</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="E5:14_2_1_2_5"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></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><a NAME="E7:14_2_1_2"/>
<font color="Maroon">S1</font> is <a href="tbsp_1.html#V3" target="_self">Cauchy</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:14_2_1_2_6"/>
for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font>, <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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<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 class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_6_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_2_1_2_6_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:14_2_1_2_6_1"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E3:14_2_1_2_6_1"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E4:14_2_1_2_6_1"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E5:14_2_1_2_6_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</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="E7:14_2_1_2_6_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<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> , <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="E8:14_2_1_2_6_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E9:14_2_1_2_6_1"/><b>then </b>
<font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:14_2_1_2_6_1"/><b>then </b>
<font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:14_2_1_2_6_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span></span>)</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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E12:14_2_1_2_6_1"/><i><font color="Green">E106</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span></span>)</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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">L</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> 0</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">L</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>
<br/>.= 
1 <a href="real_1.html#K4">*</a> <font color="Maroon">s</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">s</font>

;<br/>
<b>hence </b><a NAME="E14:14_2_1_2_6_1"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</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>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</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="E2:14_2_1_2_6"/>
<font color="Maroon">S1</font> is <a href="tbsp_1.html#V3" target="_self">Cauchy</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E8:14_2_1_2"/><b>then </b>
<font color="Maroon">S1</font> is <a href="tbsp_1.html#V2" target="_self">convergent</a>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E10:14_2_1_2"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</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/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i/>;<br/><a NAME="E11:14_2_1_2"/><i><font color="Green">E108</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_2_7"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:14_2_1_2_7"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>
<a NAME="E3:14_2_1_2_7"/><i><font color="Green">E110</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E4:14_2_1_2_7"/><b>then </b><i><font color="Green">E111</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 4
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 4;<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="E6:14_2_1_2_7"/><i><font color="Green">E112</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="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 4
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:14_2_1_2_7"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:14_2_1_2_7"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 4
 
<b>by </b><i/>;<br/>
<a NAME="E9:14_2_1_2_7"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:14_2_1_2_7"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:14_2_1_2_7"/><i><font color="Green">E115</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 4
 
<b>by </b><i/>;<br/>
<a NAME="E12:14_2_1_2_7"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E13:14_2_1_2_7"/><b>then </b>
<font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T147">REAL_2:147</a></i>;<br/>
<a NAME="E14:14_2_1_2_7"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:14_2_1_2_7"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 4
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:14_2_1_2_7"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<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></span>)</span> <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<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></span>)</span> <a href="real_1.html#K6">/</a> 4</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E17:14_2_1_2_7"/><i><font color="Green">E117</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E18:14_2_1_2_7"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<b>hence </b><a NAME="E19:14_2_1_2_7"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:14_2_1_2"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E1:14_2_1_2_8"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<b>assume </b><a NAME="E2:14_2_1_2_8"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E3:14_2_1_2_8"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>
<a NAME="E4:14_2_1_2_8"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E5:14_2_1_2_8"/><b>then </b>
<font color="Maroon">L</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T145">REAL_2:145</a></i>;<br/>
<b>hence </b><a NAME="E6:14_2_1_2_8"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E13:14_2_1_2"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:14"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> ) )
 ;<br/>


</div>
