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

<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">V</font> is <a href="vectmetr.html#V1" target="_self">convex</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E2:3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">a1</font>, <font color="Olive">a2</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">a2</font></span><a href="domain_1.html#K1">]</a></span> holds <br/> ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Olive">a2</font></span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">a1</font>,<font color="Olive">a2</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Olive">a1</font>,<font color="Olive">b</font></span>)</span> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">a1</font>,<font color="Olive">a2</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Olive">b</font>,<font color="Olive">a2</font></span>)</span> );<br/>
<a NAME="E3:3"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>  ex <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">c</font>,<font color="Olive">d</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">a1'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:3_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a1'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E3:3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a2'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E4:3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">a2'</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1'</font> = <font color="Maroon">a1'</font>, <font color="Maroon">a2'</font> = <font color="Maroon">a2'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:3_1"/><i><font color="Green">E59</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1'</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font></span>)</span>
 <b>and </b><br/><a NAME="E8:3_1"/><i><font color="Green">E60</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">a2'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</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> <font color="Maroon">a1'</font>,<font color="Maroon">a2'</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">c</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>;<br/>

<b>take </b><font color="Maroon">d</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">a2'</font></span><a href="domain_1.html#K1">]</a></span>;<br/>

<b>let </b><font color="Maroon">a1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>, <font color="Maroon">a2</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E9:3_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E10:3_1"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E11:3_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>

<b>thus </b><a NAME="E12:3_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>

<b>thus </b><a NAME="E13:3_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E14:3_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">a2</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">D</font> being  <a href="bintree1.html#V2">binary</a> <a href="trees_2.html#NM6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E5:3"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="finseq_2.html#K3">*</a> 
 <b>and </b><br/><a NAME="E6:3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E7:3"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">z</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Olive">z</font>,<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>]
 <b>from </b><i><a class="ref" href="bintree2.html#S1">BINTREE2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">dD</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font> as  <a href="bintree2.html#V1">full</a> <a href="trees_1.html#NM1">Tree</a> <b>by </b><i>, <a class="ref" href="bintree2.html#D2">BINTREE2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_2"/>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> or  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1"><b>suppose </b></a><a NAME="E1:3_2_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="real_2.html#T143">REAL_2:143</a></i>;<br/><a NAME="E3:3_2_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<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>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,1
 <b>by </b><i><a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E4:3_2_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<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>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="power.html#T59">POWER:59</a></i>;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T80">INT_1:80</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E7:3_2_1"/><i><font color="Green">E70</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><b>reconsider </b><font color="Maroon">N</font> = 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font>;<br/><a NAME="E9:3_2_1"/>
 <a href="power.html#K6">log</a> 2,<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>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T52">INT_1:52</a></i>;<br/><a NAME="E10:3_2_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<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>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E11:3_2_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<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>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E12:3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E13:3_2_1"/><b>then </b>
<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>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">N</font> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E14:3_2_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">N</font> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E15:3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E16:3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E17:3_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>reconsider </b><font color="Maroon">FSL</font> =  <a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> = <font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">FSL</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">g</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E20:3_2_1"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E21:3_2_1"/><i><font color="Green">E77</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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> = <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a> ;<br/><b>consider </b><font color="Maroon">h</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E23:3_2_1"/><i><font color="Green">E80</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E24:3_2_1"/><i><font color="Green">E81</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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><b>take </b><font color="Maroon">f</font> = <font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="binarith.html#K11">*&gt;</a></span>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>;<br/><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> 0</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>
;<br/><a NAME="E26:3_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E27:3_2_1"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_2"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">zj</font> =  <a href="euclid.html#K5">0*</a> <font color="Maroon">j</font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="binari_3.html#T5">BINARI_3:5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:3_2_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E4:3_2_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E5:3_2_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">zj</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">z1</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:3_2_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">zj</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">z1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E9:3_2_1_2"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">zj</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">b1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E10:3_2_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">z1</font></span>)</span>
 <b>and </b><br/><a NAME="E11:3_2_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">z1</font>,<font color="Maroon">b1</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E12:3_2_1_2"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">zj</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="binari_3.html#T4">BINARI_3:4</a></i>
<br/>.= 
<font color="Maroon">a1</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E28:3_2_1"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><b>defpred </b><font color="Maroon">S<sub>3</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">t</font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">a<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>   st <font color="Olive">t</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Olive">t</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>;<br/><a NAME="E29:3_2_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>reconsider </b><font color="Maroon">pusty</font> =  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:3_2_1_3"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:3_2_1_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E5:3_2_1_3"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E6:3_2_1_3"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E7:3_2_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> 1
 ;<br/>

<b>then </b><font color="Maroon">t</font> = 
1 <a href="finseqop.html#K4">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="binari_3.html#T15">BINARI_3:15</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E30:3_2_1"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">j</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3_2_1_4"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">t</font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">j</font>,<a href="margrel1.html#K6">BOOLEAN</a>   st <font color="Olive">t</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Olive">t</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<b>let </b><font color="Maroon">t</font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>assume </b><a NAME="E2:3_2_1_4"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">t1</font> being   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">j</font>,<a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">dd</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E4:3_2_1_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="binarith.html#K10">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">dd</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<a NAME="E5:3_2_1_4"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<font color="Maroon">dd</font> = 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>
;<br/>
<a NAME="E7:3_2_1_4"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <font color="Maroon">dd</font> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E8:3_2_1_4"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="binarith.html#K4">'not'</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">t1</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="binari_2.html#T11">BINARI_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">nt1</font> =  <a href="binarith.html#K4">'not'</a> <font color="Maroon">t1</font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:3_2_1_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E12:3_2_1_4"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E13:3_2_1_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">nt1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E16:3_2_1_4"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">nt1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E17:3_2_1_4"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">nt1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E18:3_2_1_4"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E19:3_2_1_4"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b</font>,<font color="Maroon">b1</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E20:3_2_1_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="binari_3.html#T4">BINARI_3:4</a></i>;<br/>
<a NAME="E21:3_2_1_4"/><b>then </b>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E22:3_2_1_4"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  = 
<font color="Maroon">b1</font>
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E31:3_2_1"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">j</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, );<br/></i><i><font color="Green">E114</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E33:3_2_1"/><i><font color="Green">E115</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E34:3_2_1"/><b>then </b><i><font color="Green">E116</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E35:3_2_1"/><b>then </b><i><font color="Green">E117</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E36:3_2_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E37:3_2_1"/><b>then </b><i><font color="Green">E118</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span>
 <b>by </b><i><a class="ref" href="bintree2.html#T19">BINTREE2:19</a></i>;<br/><a NAME="E38:3_2_1"/>
1 <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/><b>hence </b><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">FSL</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="bintree2.html#T15">BINTREE2:15</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i/>
;<br/><br/><a NAME="E40:3_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E41:3_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><i><font color="Green">E119</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><br/><a NAME="E43:3_2_1"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:3_2_1_8"/><i><font color="Green">E121</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:3_2_1_8"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/>

<a NAME="E3:3_2_1_8"/><i><font color="Green">E123</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FSL</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="bintree2.html#T19">BINTREE2:19</a></i>;<br/>
<a NAME="E4:3_2_1_8"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:3_2_1_8"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E125</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, , , , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">FSL</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, , , , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/>
<a NAME="E7:3_2_1_8"/><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:3_2_1_8"/>
 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="binari_3.html#T5">BINARI_3:5</a></i>;<br/>
<a NAME="E9:3_2_1_8"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E10:3_2_1_8"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T8">JORDAN2B:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ze</font> =  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<div><i><font color="Green">E129</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_2_1_8_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_2_1_1"><b>suppose </b></a><a NAME="E1:3_2_1_8_2_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><div class="add"><a NAME="E2:3_2_1_8_2_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E3:3_2_1_8_2_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:3_2_1_8_2_1_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E5:3_2_1_8_2_1_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[ non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> ;<br/><a NAME="E6:3_2_1_8_2_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8_2_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:3_2_1_8_2_1_1_1"/><i><font color="Green">E134</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:3_2_1_8_2_1_1_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/>

<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
1

;<br/>
<a NAME="E4:3_2_1_8_2_1_1_1"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pusty</font> =  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:3_2_1_8_2_1_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E8:3_2_1_8_2_1_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E9:3_2_1_8_2_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E10:3_2_1_8_2_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> 1,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="bintree2.html#T23">BINTREE2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="vectmetr.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> 1,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="bintree2.html#T22">BINTREE2:22</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E7:3_2_1_8_2_1_1"/><i><font color="Green">E139</font></i>: 
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>4</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="3_2_1_8_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3_2_1_8_2_1_1_2"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 ;<br/>

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

<b>assume </b><b>that </b><br/><a NAME="E2:3_2_1_8_2_1_1_2"/><i><font color="Green">E141</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E3:3_2_1_8_2_1_1_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">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#K5">-</a> 1
 ;<br/>

<a NAME="E4:3_2_1_8_2_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:3_2_1_8_2_1_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:3_2_1_8_2_1_1_2"/>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E7:3_2_1_8_2_1_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:3_2_1_8_2_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">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="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="vectmetr.html#T1" target="_self">Th1</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E9:3_2_1_8_2_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:3_2_1_8_2_1_1_2"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<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#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E11:3_2_1_8_2_1_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:3_2_1_8_2_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>
<a NAME="E13:3_2_1_8_2_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E14:3_2_1_8_2_1_1_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_2.html#T27">NAT_2:27</a></i>;<br/>
<a NAME="E15:3_2_1_8_2_1_1_2"/><i><font color="Green">E148</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E16:3_2_1_8_2_1_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T15">NAT_2:15</a></i>;<br/>
<a NAME="E17:3_2_1_8_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E18:3_2_1_8_2_1_1_2"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20">BINTREE2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">zb</font> = <font color="Maroon">dD</font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">n</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="bintree2.html#T10">BINTREE2:10</a></i>;<br/>
<a NAME="E20:3_2_1_8_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">zb</font>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E3:3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="bintree2.html#T5">BINTREE2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E23:3_2_1_8_2_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>
<a NAME="E24:3_2_1_8_2_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E25:3_2_1_8_2_1_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_2.html#T27">NAT_2:27</a></i>;<br/>
<a NAME="E26:3_2_1_8_2_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="E27:3_2_1_8_2_1_1_2"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E28:3_2_1_8_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T15">NAT_2:15</a></i>;<br/>
<a NAME="E29:3_2_1_8_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E30:3_2_1_8_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20">BINTREE2:20</a></i>;<br/>
<a NAME="E31:3_2_1_8_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">zb</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">FF</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="bintree2.html#T5">BINTREE2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">FF1</font> = <font color="Maroon">FF</font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E35:3_2_1_8_2_1_1_2"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E36:3_2_1_8_2_1_1_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E37:3_2_1_8_2_1_1_2"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">a'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E39:3_2_1_8_2_1_1_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E40:3_2_1_8_2_1_1_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E41:3_2_1_8_2_1_1_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">FF1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font>, <font color="Maroon">a1'</font> = <font color="Maroon">a'</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E44:3_2_1_8_2_1_1_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">d</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E45:3_2_1_8_2_1_1_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">b1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E46:3_2_1_8_2_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">d</font></span>)</span>
 <b>and </b><br/><a NAME="E47:3_2_1_8_2_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">d</font>,<font color="Maroon">b1</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">d'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E49:3_2_1_8_2_1_1_2"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">FF1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">d'</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E50:3_2_1_8_2_1_1_2"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">FF1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d'</font>,<font color="Maroon">b1'</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E51:3_2_1_8_2_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">a1'</font>,<font color="Maroon">d'</font></span>)</span>
 <b>and </b><br/><a NAME="E52:3_2_1_8_2_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">d'</font>,<font color="Maroon">b1'</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E53:3_2_1_8_2_1_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E54:3_2_1_8_2_1_1_2"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<i><font color="Green">E172</font></i>: <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8_2_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_2_1_8_2_1_1_2_2_1"/>
(  not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">i</font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:3_2_1_8_2_1_1_2_2_1_1"/>
not <font color="Maroon">i</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><a NAME="E2:3_2_1_8_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="nat_2.html#T24">NAT_2:24</a></i>;<br/><a NAME="E3:3_2_1_8_2_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="vectmetr.html#E5:3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_2.html#T20">NAT_2:20</a></i>;<br/><a NAME="E4:3_2_1_8_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="ref" href="nat_2.html#T23">NAT_2:23</a></i>;<br/><a NAME="E5:3_2_1_8_2_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T28">NAT_2:28</a></i>;<br/><a NAME="E6:3_2_1_8_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E174</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">FF</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="bintree2.html#T24">BINTREE2:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">FF</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E6:3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
<br/>.= 
<font color="Maroon">d</font>
<b>by </b><i>, <a class="txt" href="vectmetr.html#E7:3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="txt" href="vectmetr.html#E5:3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="vectmetr.html#E6:3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_2.html#T20">NAT_2:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="bintree2.html#T24">BINTREE2:24</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:3_2_1_8_2_1_1_2_2_1_2"/>
<font color="Maroon">i</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><a NAME="E2:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_2.html#T23">NAT_2:23</a></i>;<br/><b>then </b><i><font color="Green">E176</font></i>: 1 = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_2.html#T20">NAT_2:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/><a NAME="E4:3_2_1_8_2_1_1_2_2_1_2"/><i><font color="Green">E177</font></i>: 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E6:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E179</font></i>: <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> 1

;<br/><a NAME="E8:3_2_1_8_2_1_1_2_2_1_2"/>
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3"><i><font color="Green">E66</font></i></a>, , , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E9:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="nat_2.html#T20">NAT_2:20</a></i>;<br/><a NAME="E10:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><a NAME="E11:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E180</font></i>: 
2 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/><a NAME="E12:3_2_1_8_2_1_1_2_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="E13:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_2.html#T17">NAT_2:17</a></i>;<br/><a NAME="E14:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E15:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2
 ;<br/><a NAME="E16:3_2_1_8_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E182</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><i><font color="Green">E183</font></i>: <font color="Maroon">a'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, , , <a class="txt" href="vectmetr.html#E17:3_2_1"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">FF</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="bintree2.html#T24">BINTREE2:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">FF</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E6:3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
<br/>.= 
<font color="Maroon">a1'</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3"><i><font color="Green">E43</font></i></a>, , <a class="txt" href="vectmetr.html#E7:3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="bintree2.html#T24">BINTREE2:24</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E57:3_2_1_8_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E8:3_2_1_8_2_1_1"/><i><font color="Green">E184</font></i>: 
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, );<br/></i><a NAME="E9:3_2_1_8_2_1_1"/><i><font color="Green">E185</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E10:3_2_1_8_2_1_1"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:3_2_1_8_2_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i>, , <a class="ref" href="vectmetr.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, , , , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">FSL</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i>, , <a class="ref" href="vectmetr.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, , , , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_2_1_2"><b>suppose </b></a><a NAME="E1:3_2_1_8_2_1_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><div class="add"><b>hence </b><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">ze</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">n</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, , , <a class="ref" href="bintree2.html#T16">BINTREE2:16</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:3_2_1_8"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E14:3_2_1_8"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[ non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">j</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">DF1</font>, <font color="Olive">DF2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">DF1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Olive">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">DF1</font>,<font color="Olive">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E15:3_2_1_8"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:3_2_1_8_3"/><i><font color="Green">E193</font></i>: 
2 <a href="series_1.html#K3">to_power</a> 1 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E2:3_2_1_8_3"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>assume </b><a NAME="E3:3_2_1_8_3"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> 1
 ;<br/>

<a NAME="E4:3_2_1_8_3"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>let </b><font color="Maroon">DF1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">DF2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E5:3_2_1_8_3"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">DF1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> 1,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>and </b><br/><a NAME="E6:3_2_1_8_3"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> 1,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">pusty</font> =  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E8:3_2_1_8_3"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E9:3_2_1_8_3"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:3_2_1_8_3"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> 1,<font color="Maroon">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20">BINTREE2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">FSL1j</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> 1,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_2_1_8_3_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_3_1_1_1"><b>suppose </b></a><a NAME="E1:3_2_1_8_3_1_1_1"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><i><font color="Green">E201</font></i>: <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> = 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">FSL1j</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="bintree2.html#T22">BINTREE2:22</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">DF1</font>,<font color="Maroon">DF2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="vectmetr.html#T1" target="_self">Th1</a>, <a class="txt" href="vectmetr.html#E1:3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>
;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:3_2_1_8_3_1_1_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E5:3_2_1_8_3_1_1_1"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E6:3_2_1_8_3_1_1_1"/><i><font color="Green">E203</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E7:3_2_1_8_3_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:3_2_1_8_3_1_1_1"/>
( <font color="Maroon">DF1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E9:3_2_1_8_3_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">DF1</font>,<font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_3_1_1_2"><b>suppose </b></a><a NAME="E1:3_2_1_8_3_1_1_2"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><i><font color="Green">E205</font></i>: <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> = 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">FSL1j</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="bintree2.html#T23">BINTREE2:23</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">DF1</font>,<font color="Maroon">DF2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="vectmetr.html#T1" target="_self">Th1</a>, <a class="txt" href="vectmetr.html#E1:3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>
;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:3_2_1_8_3_1_1_2"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E5:3_2_1_8_3_1_1_2"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pusty</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E6:3_2_1_8_3_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E7:3_2_1_8_3_1_1_2"/><i><font color="Green">E207</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:3_2_1_8_3_1_1_2"/>
( <font color="Maroon">DF1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E9:3_2_1_8_3_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">DF1</font>,<font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2
 <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="E13:3_2_1_8_3"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">DF1</font>,<font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:3_2_1_8"/><i><font color="Green">E208</font></i>: 
for <font color="Olive">l</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">l</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3_2_1_8_4"/><i><font color="Green">E210</font></i>: 
for <font color="Olive">j</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font> holds <br/>for <font color="Olive">DF1</font>, <font color="Olive">DF2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">DF1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">l</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Olive">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">l</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">DF1</font>,<font color="Olive">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">j</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:3_2_1_8_4"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">DF1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">DF2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:3_2_1_8_4"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">DF1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>and </b><br/><a NAME="E4:3_2_1_8_4"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 ;<br/>

<a NAME="E5:3_2_1_8_4"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_2_1_8_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:3_2_1_8_4_1"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T14">NAT_2:14</a></i>;<br/>
<a NAME="E3:3_2_1_8_4_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E4:3_2_1_8_4_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">j1</font> = <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E7:3_2_1_8_4"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E8:3_2_1_8_4"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>
<a NAME="E9:3_2_1_8_4"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K2">*</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E10:3_2_1_8_4"/><b>then </b><i><font color="Green">E216</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="nat_2.html#T27">NAT_2:27</a></i>;<br/>
<a NAME="E11:3_2_1_8_4"/><b>then </b>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:3_2_1_8_4"/><b>then </b><i><font color="Green">E217</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">l</font>,<font color="Maroon">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20">BINTREE2:20</a></i>;<br/>
<a NAME="E13:3_2_1_8_4"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E14:3_2_1_8_4"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:3_2_1_8_4"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20">BINTREE2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">dDll1</font> = <font color="Maroon">dD</font> <a href="trees_2.html#K2">-level</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="bintree2.html#T10">BINTREE2:10</a></i>;<br/>
<a NAME="E17:3_2_1_8_4"/>
<span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dDll1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">FSLl1</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="bintree2.html#T5">BINTREE2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Fj</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E20:3_2_1_8_4"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">Fj</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">DF1</font>,<font color="Maroon">DF2</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, <a class="ref" href="vectmetr.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">FSLl</font> being   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">l</font>,<a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E22:3_2_1_8_4"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">FSLl1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSLl</font> <a href="binarith.html#K10">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<b>reconsider </b><font color="Maroon">NFSLl</font> = <font color="Maroon">FSLl</font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E25:3_2_1_8_4"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E26:3_2_1_8_4"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E27:3_2_1_8_4"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">D</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">NFSLl</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">y1</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="txt" href="vectmetr.html#E3:3"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E30:3_2_1_8_4"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">NFSLl</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E31:3_2_1_8_4"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">NFSLl</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E32:3_2_1_8_4"/><i><font color="Green">E233</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E33:3_2_1_8_4"/><i><font color="Green">E234</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b</font>,<font color="Maroon">y1</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">dDll</font> = <font color="Maroon">dD</font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">l</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="bintree2.html#T10">BINTREE2:10</a></i>;<br/>
<a NAME="E35:3_2_1_8_4"/>
<span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">l</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">dDll</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">FSLlprim</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">l</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">l</font>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="bintree2.html#T5">BINTREE2:5</a></i>;<br/>
<a NAME="E37:3_2_1_8_4"/>
<span class="p1">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSLlprim</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="bintree2.html#T24">BINTREE2:24</a></i>;<br/>
<a NAME="E38:3_2_1_8_4"/><b>then </b><i><font color="Green">E237</font></i>: 
( <font color="Maroon">FSLl</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSLlprim</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2 )
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E39:3_2_1_8_4"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">l</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2">FinSeqLevel</a> <font color="Maroon">l</font>,<font color="Maroon">dD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E40:3_2_1_8_4"/><b>then </b><i><font color="Green">E238</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="vectmetr.html#E2:3"><i><font color="Green">E44</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8_4_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_2_1_8_4_2_1"/>
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T62">NAT_1:62</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_4_2_1_1"><b>suppose </b></a><a NAME="E1:3_2_1_8_4_2_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:3_2_1_8_4_2_1_1"/><b>then </b>
( <font color="Maroon">DF1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E3:3_2_1_8_4_2_1_1"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">DF1</font>,<font color="Maroon">DF2</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i>, </i>;<br/><b>hence </b> <a href="metric_1.html#K4">dist</a> <font color="Maroon">DF1</font>,<font color="Maroon">DF2</font> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K2">*</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">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> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">l</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/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_8_4_2_1_2"><b>suppose </b></a><a NAME="E1:3_2_1_8_4_2_1_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:3_2_1_8_4_2_1_2"/><b>then </b>
( <font color="Maroon">DF1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E3:3_2_1_8_4_2_1_2"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">DF1</font>,<font color="Maroon">DF2</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i>, </i>;<br/><b>hence </b> <a href="metric_1.html#K4">dist</a> <font color="Maroon">DF1</font>,<font color="Maroon">DF2</font> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K2">*</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T79">XCMPLX_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">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> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">l</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/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E42:3_2_1_8_4"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">DF1</font>,<font color="Maroon">DF2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:3_2_1_8"/>
for <font color="Olive">l</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">l</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E18:3_2_1_8"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font>
 <b>by </b><i>, , , <a class="ref" href="vectmetr.html#D1" target="_self">Def1</a>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E44:3_2_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E45:3_2_1"/><i><font color="Green">E239</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>and </b><br/><a NAME="E46:3_2_1"/><i><font color="Green">E240</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <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="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E47:3_2_1"/><i><font color="Green">E241</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E48:3_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_9"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_2_1_9"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:3_2_1_9_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:3_2_1_9_1"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:3_2_1_9_1"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:3_2_1_9_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:3_2_1_9_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:3_2_1_9_1"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E9:3_2_1_9_1"/><i><font color="Green">E246</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c1</font>
 <b>and </b><br/><a NAME="E10:3_2_1_9_1"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 ;<br/>
<font color="Maroon">a</font> = 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c1</font>
<b>by </b><i><a class="ref" href="vectmetr.html#D1" target="_self">Def1</a>, , , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font>
<b>by </b><i>, , , </i>
;<br/>
<b>hence </b><a NAME="E12:3_2_1_9_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:3_2_1_9"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:3_2_1_9"/><b>then </b><i><font color="Green">E248</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:3_2_1_9"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:3_2_1_9"/><b>then </b><i><font color="Green">E249</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<font color="Maroon">a</font> = 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/>
<b>hence </b><a NAME="E7:3_2_1_9"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="vectmetr.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><b>then </b><font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</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>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T15">FUNCOP_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</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>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#D2">FINSEQ_2:def 2</a></i>
;<br/><b>hence </b> <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</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> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>
<br/>.= 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_2"><b>suppose </b></a><a NAME="E1:3_2_2"/><i><font color="Green">E250</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">f</font> = <span class="p1"><a href="rlvect_1.html#K7">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="rlvect_1.html#K7">*&gt;</a></span>;<br/><b>thus </b><a NAME="E2:3_2_2"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><a NAME="E3:3_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E4:3_2_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><i><font color="Green">E252</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
1

;<br/><b>thus </b><a NAME="E6:3_2_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:3_2_2_2"/><i><font color="Green">E253</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:3_2_2_2"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/>

<a NAME="E3:3_2_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:3_2_2_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:3_2_2_2"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E7:3_2_2"/><i><font color="Green">E255</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>and </b><br/><a NAME="E8:3_2_2"/><i><font color="Green">E256</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <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="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 = 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
;<br/><a NAME="E10:3_2_2"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_5.html#T15">FINSEQ_5:15</a></i>;<br/><b>hence </b><a NAME="E11:3_2_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T103">RVSUM_1:103</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
