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

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c1">V</font> is <a href="vectmetr.html#V1" title="VECTMETR:attr.1">convex</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E2:3"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c5">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/> ex <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">b</font></span>)</span> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">a2</font></span>)</span> );<br/>
<a NAME="E3:3"/><i><font color="Green" title="E5">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  ex <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">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" title="c6">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>consider </b><font color="Maroon" title="c7">a1'</font>, <font color="Maroon" title="c8">a2'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:3_1"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c7">a1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E3:3_1"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c8">a2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E4:3_1"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a1'</font>,<font color="Maroon" title="c8">a2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a1'</font> = <font color="Maroon" title="c7">a1'</font>, <font color="Maroon" title="c10">a2'</font> = <font color="Maroon" title="c8">a2'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="vectmetr.html#E2:3_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="vectmetr.html#E3:3_1"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:3_1"/><i><font color="Green" title="E9">A7</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c10">a2'</font></span>)</span>
 <b>and </b><br/><a NAME="E8:3_1"/><i><font color="Green" title="E10">A8</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c10">a2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c10">a2'</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="vectmetr.html#D1" target="_self" title="VECTMETR:def.1">Def1</a></i>;<br/>
<b>take </b><font color="Maroon" title="c12">c</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c11">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>

<b>take </b><font color="Maroon" title="c13">d</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c10">a2'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>

<b>let </b><font color="Maroon" title="c14">a1</font>, <font color="Maroon" title="c15">a2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E9:3_1"/><i><font color="Green" title="E11">A9</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/>

<a NAME="E10:3_1"/><b>then </b><i><font color="Green" title="E12">A10</font></i>: 
( <font color="Maroon" title="c14">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1'</font> &amp; <font color="Maroon" title="c15">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a2'</font> )
 
<b>by </b><i><a class="txt" href="vectmetr.html#E4:3_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">z</font>
;<br/>

<b>thus </b><a NAME="E11:3_1"/>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c11">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E4:3_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="vectmetr.html#E9:3_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>

<b>thus </b><a NAME="E12:3_1"/>
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c15">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E4:3_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="vectmetr.html#E9:3_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>

<b>thus </b><a NAME="E13:3_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c11">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="vectmetr.html#E10:3_1"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:3_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">a1</font>,<font color="Maroon" title="c15">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c15">a2</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E8:3_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="vectmetr.html#E10:3_1"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">D</font> being  <a href="bintree1.html#V2" title="BINTREE1:attr.2">binary</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E5:3"/><i><font color="Green" title="E6">A11</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <b>and </b><br/><a NAME="E6:3"/><i><font color="Green" title="E7">A12</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E7:3"/><i><font color="Green" title="E8">A13</font></i>: 
for <font color="Olive" title="b1">z</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">D</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">z</font>,<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>]
 <b>from </b><i><a class="ref" href="bintree2.html#S1" title="BINTREE2:sch.1">BINTREE2:sch 1</a>(<a class="txt" href="vectmetr.html#E3:3"><i><font color="Green" title="E5">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c7">dD</font> =  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">D</font> as  <a href="bintree2.html#V1" title="BINTREE2:attr.1">full</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="bintree2.html#D2" title="BINTREE2:def.2">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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">p</font> or  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="real_2.html#T143" title="REAL_2:th.143">REAL_2:143</a></i>;<br/><a NAME="E3:3_2_1"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,1
 <b>by </b><i><a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></i>;<br/><a NAME="E4:3_2_1"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">n1</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T80" title="INT_1:th.80">INT_1:80</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c8">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E7:3_2_1"/><i><font color="Green" title="E9">A14</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">N</font> = 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c9">n</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><b>set </b><font color="Maroon" title="c11">r</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font>;<br/><a NAME="E9:3_2_1"/>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T52" title="INT_1:th.52">INT_1:52</a></i>;<br/><a NAME="E10:3_2_1"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> 2,2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T60" title="POWER:th.60">POWER:60</a></i>;<br/><a NAME="E11:3_2_1"/><b>then </b>
 <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63" title="POWER:th.63">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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">N</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E14:3_2_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">N</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">N</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>;<br/><a NAME="E17:3_2_1"/><b>then </b><i><font color="Green" title="E10">A15</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">FSL</font> =  <a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c7">dD</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">D</font> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = <font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">FSL</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c13">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E20:3_2_1"/><i><font color="Green" title="E11">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">N</font>
 <b>and </b><br/><a NAME="E21:3_2_1"/><i><font color="Green" title="E12">A17</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">N</font> holds <br/><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> ;<br/><b>consider </b><font color="Maroon" title="c14">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E23:3_2_1"/><i><font color="Green" title="E13">A18</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">N</font>
 <b>and </b><br/><a NAME="E24:3_2_1"/><i><font color="Green" title="E14">A19</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">N</font> holds <br/><font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><b>take </b><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c14">h</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> $1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>;<br/><a NAME="E25:3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 <b>by </b><i><a class="txt" href="vectmetr.html#E6:3"><i><font color="Green" title="E7">A12</font></i></a>, <a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>;<br/><a NAME="E26:3_2_1"/><b>then </b><i><font color="Green" title="E15">A20</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E27:3_2_1"/><i><font color="Green" title="E16">A21</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c17">zj</font> =  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c16">j</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="binari_3.html#T5" title="BINARI_3:th.5">BINARI_3:5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">a</font>, <font color="Maroon" title="c19">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_2_1_1"/><i><font color="Green" title="E17">A22</font></i>: 
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E4:3_2_1_1"/><i><font color="Green" title="E18">A23</font></i>: 
<font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E5:3_2_1_1"/><i><font color="Green" title="E19">A24</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c17">zj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c19">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">a1</font> = <font color="Maroon" title="c18">a</font>, <font color="Maroon" title="c21">b1</font> = <font color="Maroon" title="c19">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="vectmetr.html#E3:3_2_1_1"><i><font color="Green" title="E17">A22</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_1"><i><font color="Green" title="E18">A23</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">z1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E8:3_2_1_1"/><i><font color="Green" title="E20">A25</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">zj</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c22">z1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E9:3_2_1_1"/>
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">zj</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">z1</font>,<font color="Maroon" title="c21">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E10:3_2_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c22">z1</font></span>)</span>
 <b>and </b><br/><a NAME="E11:3_2_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c22">z1</font>,<font color="Maroon" title="c21">b1</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="vectmetr.html#E5:3_2_1_1"><i><font color="Green" title="E19">A24</font></i></a></i>;<br/>
<b>assume </b><a NAME="E12:3_2_1_1"/><i><font color="Green" title="E21">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">zj</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</a></i>
<br/>.= 
<font color="Maroon" title="c20">a1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E8:3_2_1_1"><i><font color="Green" title="E20">A25</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E5:3_2_1_1"><i><font color="Green" title="E19">A24</font></i></a>, <a class="txt" href="vectmetr.html#E12:3_2_1_1"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E28:3_2_1"/><i><font color="Green" title="E17">A27</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="vectmetr.html#E26:3_2_1"><i><font color="Green" title="E15">A20</font></i></a>, <a class="txt" href="vectmetr.html#E27:3_2_1"><i><font color="Green" title="E16">A21</font></i></a>);<br/></i><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">t</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of $1,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> $1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>;<br/><a NAME="E29:3_2_1"/><i><font color="Green" title="E18">A28</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[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" title="c16">t</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of 1,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c17">pusty</font> =  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:3_2_1_2"/>
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c18">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:3_2_1_2"/><i><font color="Green" title="E19">A29</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E5:3_2_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c18">b</font></span>)</span>
 <b>and </b><br/><a NAME="E6:3_2_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E6:3"><i><font color="Green" title="E7">A12</font></i></a>, <a class="txt" href="vectmetr.html#E7:3"><i><font color="Green" title="E8">A13</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:3_2_1_2"/>
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 1
 ;<br/>

<a NAME="E8:3_2_1_2"/><b>then </b>
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T73" title="FINSEQ_2:th.73">FINSEQ_2:73</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c16">t</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="binari_3.html#T15" title="BINARI_3:th.15">BINARI_3:15</a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:3_2_1_2"><i><font color="Green" title="E19">A29</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E30:3_2_1"/><i><font color="Green" title="E19">A30</font></i>: 
for <font color="Olive" title="b1">j</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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" title="c16">j</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3_2_1_3"/><i><font color="Green" title="E20">A31</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c16">j</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c16">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c17">t</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/>

<b>assume </b><a NAME="E2:3_2_1_3"/><i><font color="Green" title="E21">A32</font></i>: 
<font color="Maroon" title="c17">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c18">t1</font> being   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c16">j</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> , <font color="Maroon" title="c19">dd</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E4:3_2_1_3"/><i><font color="Green" title="E22">A33</font></i>: 
<font color="Maroon" title="c17">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c19">dd</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137" title="FINSEQ_2:th.137">FINSEQ_2:137</a></i>;<br/>
<a NAME="E5:3_2_1_3"/><i><font color="Green" title="E23">A34</font></i>: 
<font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<font color="Maroon" title="c19">dd</font> = 
<font color="Maroon" title="c17">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">t1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:3_2_1_3"><i><font color="Green" title="E22">A33</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_3"><i><font color="Green" title="E21">A32</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E5:3_2_1_3"><i><font color="Green" title="E23">A34</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E7:3_2_1_3"/><b>then </b>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c19">dd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<a NAME="E8:3_2_1_3"/><b>then </b><i><font color="Green" title="E24">A35</font></i>: 
 <a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c17">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c18">t1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E4:3_2_1_3"><i><font color="Green" title="E22">A33</font></i></a>, <a class="ref" href="binari_2.html#T11" title="BINARI_2:th.11">BINARI_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">nt1</font> =  <a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c18">t1</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">a</font>, <font color="Maroon" title="c22">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:3_2_1_3"/><i><font color="Green" title="E25">A36</font></i>: 
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E12:3_2_1_3"/><i><font color="Green" title="E26">A37</font></i>: 
<font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E13:3_2_1_3"/><i><font color="Green" title="E27">A38</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c20">nt1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">a</font>,<font color="Maroon" title="c22">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">a1</font> = <font color="Maroon" title="c21">a</font>, <font color="Maroon" title="c24">b1</font> = <font color="Maroon" title="c22">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="vectmetr.html#E11:3_2_1_3"><i><font color="Green" title="E25">A36</font></i></a>, <a class="txt" href="vectmetr.html#E12:3_2_1_3"><i><font color="Green" title="E26">A37</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E16:3_2_1_3"/>
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">nt1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c23">a1</font>,<font color="Maroon" title="c25">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E17:3_2_1_3"/><i><font color="Green" title="E28">A39</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">nt1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">b</font>,<font color="Maroon" title="c24">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E18:3_2_1_3"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">a1</font>,<font color="Maroon" title="c24">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">a1</font>,<font color="Maroon" title="c25">b</font></span>)</span>
 <b>and </b><br/><a NAME="E19:3_2_1_3"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">a1</font>,<font color="Maroon" title="c24">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">b</font>,<font color="Maroon" title="c24">b1</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="vectmetr.html#E13:3_2_1_3"><i><font color="Green" title="E27">A38</font></i></a></i>;<br/>
<a NAME="E20:3_2_1_3"/>
<font color="Maroon" title="c17">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_3"><i><font color="Green" title="E21">A32</font></i></a>, <a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</a></i>;<br/>
<a NAME="E21:3_2_1_3"/><b>then </b>
<font color="Maroon" title="c18">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c16">j</font>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E4:3_2_1_3"><i><font color="Green" title="E22">A33</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E22:3_2_1_3"/><b>then </b><i><font color="Green" title="E29">A40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c18">t1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_3"><i><font color="Green" title="E20">A31</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<font color="Maroon" title="c24">b1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E8:3_2_1_3"><i><font color="Green" title="E24">A35</font></i></a>, <a class="txt" href="vectmetr.html#E17:3_2_1_3"><i><font color="Green" title="E28">A39</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E13:3_2_1_3"><i><font color="Green" title="E27">A38</font></i></a>, <a class="txt" href="vectmetr.html#E22:3_2_1_3"><i><font color="Green" title="E29">A40</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E31:3_2_1"/><i><font color="Green" title="E20">A41</font></i>: 
for <font color="Olive" title="b1">j</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="vectmetr.html#E29:3_2_1"><i><font color="Green" title="E18">A28</font></i></a>, <a class="txt" href="vectmetr.html#E30:3_2_1"><i><font color="Green" title="E19">A30</font></i></a>);<br/></i><i><font color="Green" title="E21">A42</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/><a NAME="E33:3_2_1"/><i><font color="Green" title="E22">A43</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E34:3_2_1"/><b>then </b><i><font color="Green" title="E23">A44</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E35:3_2_1"/><b>then </b><i><font color="Green" title="E24">A45</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">h</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E36:3_2_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3_2_1"><i><font color="Green" title="E9">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E37:3_2_1"/><b>then </b><i><font color="Green" title="E25">A46</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span>
 <b>by </b><i><a class="ref" href="bintree2.html#T19" title="BINTREE2:th.19">BINTREE2:19</a></i>;<br/><a NAME="E38:3_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="vectmetr.html#E35:3_2_1"><i><font color="Green" title="E24">A45</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E24:3_2_1"><i><font color="Green" title="E14">A19</font></i></a>, <a class="txt" href="vectmetr.html#E34:3_2_1"><i><font color="Green" title="E23">A44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E20:3_2_1"><i><font color="Green" title="E11">A16</font></i></a>, <a class="txt" href="vectmetr.html#E33:3_2_1"><i><font color="Green" title="E22">A43</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">FSL</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E21:3_2_1"><i><font color="Green" title="E12">A17</font></i></a>, <a class="txt" href="vectmetr.html#E34:3_2_1"><i><font color="Green" title="E23">A44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E37:3_2_1"><i><font color="Green" title="E25">A46</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="bintree2.html#T15" title="BINTREE2:th.15">BINTREE2:15</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E28:3_2_1"><i><font color="Green" title="E17">A27</font></i></a></i>
;<br/><br/><a NAME="E40:3_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/><a NAME="E41:3_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><i><font color="Green" title="E26">A47</font></i>: <font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">h</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/><br/><a NAME="E43:3_2_1"/><i><font color="Green" title="E27">A48</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:3_2_1_7"/><i><font color="Green" title="E28">A49</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font>
 <b>and </b><br/><a NAME="E2:3_2_1_7"/><i><font color="Green" title="E29">A50</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ;<br/>

<a NAME="E3:3_2_1_7"/><i><font color="Green" title="E30">A51</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">FSL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">N</font>
 
<b>by </b><i><a class="ref" href="bintree2.html#T19" title="BINTREE2:th.19">BINTREE2:19</a></i>;<br/>
<a NAME="E4:3_2_1_7"/><i><font color="Green" title="E31">A52</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E1:3_2_1_7"><i><font color="Green" title="E28">A49</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_1_7"><i><font color="Green" title="E29">A50</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:3_2_1_7"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">h</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E32">A53</font></i>: <font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> = 
<font color="Maroon" title="c14">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E1:3_2_1_7"><i><font color="Green" title="E28">A49</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_1_7"><i><font color="Green" title="E29">A50</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E24:3_2_1"><i><font color="Green" title="E14">A19</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_7"><i><font color="Green" title="E31">A52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E20:3_2_1"><i><font color="Green" title="E11">A16</font></i></a>, <a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E1:3_2_1_7"><i><font color="Green" title="E28">A49</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_1_7"><i><font color="Green" title="E29">A50</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">FSL</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E21:3_2_1"><i><font color="Green" title="E12">A17</font></i></a>, <a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_7"><i><font color="Green" title="E31">A52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E1:3_2_1_7"><i><font color="Green" title="E28">A49</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_1_7"><i><font color="Green" title="E29">A50</font></i></a>, <a class="txt" href="vectmetr.html#E3:3_2_1_7"><i><font color="Green" title="E30">A51</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/>
<a NAME="E7:3_2_1_7"/><i><font color="Green" title="E33">A54</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E8:3_2_1_7"/>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="ref" href="binari_3.html#T5" title="BINARI_3:th.5">BINARI_3:5</a></i>;<br/>
<a NAME="E9:3_2_1_7"/><b>then </b><i><font color="Green" title="E34">A55</font></i>: 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c9">n</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E10:3_2_1_7"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">ze</font> =  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c9">n</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c9">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <b>by </b><i><a class="txt" href="vectmetr.html#E9:3_2_1_7"><i><font color="Green" title="E34">A55</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<div><i><font color="Green" title="E35">A56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_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_7_2_1"/>
( <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 or <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7"><i><font color="Green" title="E29">A50</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_2_1_1"><b>suppose </b></a><a NAME="E1:3_2_1_7_2_1_1"/>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ;<br/><div class="add"><a NAME="E2:3_2_1_7_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3_2_1_7"><i><font color="Green" title="E33">A54</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E3:3_2_1_7_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:3_2_1_7_2_1_1"/><b>then </b><i><font color="Green" title="E36">A57</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3_2_1_7"><i><font color="Green" title="E33">A54</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E5:3_2_1_7_2_1_1"/><b>then </b><i><font color="Green" title="E37">A58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[ non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> $1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> $1,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> $1,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ;<br/><a NAME="E6:3_2_1_7_2_1_1"/><i><font color="Green" title="E38">A59</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:3_2_1_7_2_1_1_1"/><i><font color="Green" title="E39">A60</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font>
 <b>and </b><br/><a NAME="E2:3_2_1_7_2_1_1_1"/><i><font color="Green" title="E40">A61</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/>

<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
1

;<br/>
<a NAME="E4:3_2_1_7_2_1_1_1"/><b>then </b><i><font color="Green" title="E41">A62</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_7_2_1_1_1"><i><font color="Green" title="E39">A60</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_1"><i><font color="Green" title="E40">A61</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">pusty</font> =  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:3_2_1_7_2_1_1_1"/><i><font color="Green" title="E42">A63</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c20">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E8:3_2_1_7_2_1_1_1"/><i><font color="Green" title="E43">A64</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">b</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E9:3_2_1_7_2_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c20">b</font></span>)</span>
 <b>and </b><br/><a NAME="E10:3_2_1_7_2_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">b</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E6:3"><i><font color="Green" title="E7">A12</font></i></a>, <a class="txt" href="vectmetr.html#E7:3"><i><font color="Green" title="E8">A13</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> 1,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E4:3_2_1_7_2_1_1_1"><i><font color="Green" title="E41">A62</font></i></a>, <a class="ref" href="bintree2.html#T23" title="BINTREE2:th.23">BINTREE2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c20">b</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E8:3_2_1_7_2_1_1_1"><i><font color="Green" title="E43">A64</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E7:3_2_1_7_2_1_1_1"><i><font color="Green" title="E42">A63</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> 1,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E4:3_2_1_7_2_1_1_1"><i><font color="Green" title="E41">A62</font></i></a>, <a class="ref" href="bintree2.html#T22" title="BINTREE2:th.22">BINTREE2:22</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E7:3_2_1_7_2_1_1"/><i><font color="Green" title="E39">A65</font></i>: 
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E40">A66</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c19">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E41">A67</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font>
 <b>and </b><br/><a NAME="E3:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E42">A68</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/>

<a NAME="E4:3_2_1_7_2_1_1_2"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/>
<a NAME="E5:3_2_1_7_2_1_1_2"/><b>then </b><i><font color="Green" title="E43">A69</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:3_2_1_7_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="vectmetr.html#E3:3_2_1_7_2_1_1_2"><i><font color="Green" title="E42">A68</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E7:3_2_1_7_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:3_2_1_7_2_1_1_2"/><b>then </b><i><font color="Green" title="E44">A70</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="txt" href="vectmetr.html#E5:3_2_1_7_2_1_1_2"><i><font color="Green" title="E43">A69</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E9:3_2_1_7_2_1_1_2"/><b>then </b><i><font color="Green" title="E45">A71</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E10:3_2_1_7_2_1_1_2"/>
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E8:3_2_1_7_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>;<br/>
<a NAME="E11:3_2_1_7_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/>
<a NAME="E12:3_2_1_7_2_1_1_2"/><b>then </b><i><font color="Green" title="E46">A72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font>
 
<b>by </b><i><a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></i>;<br/>
<a NAME="E13:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E47">A73</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_2"><i><font color="Green" title="E41">A67</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:3_2_1_7_2_1_1_2"/><b>then </b><i><font color="Green" title="E48">A74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T15" title="NAT_2:th.15">NAT_2:15</a></i>;<br/>
<a NAME="E15:3_2_1_7_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E12:3_2_1_7_2_1_1_2"><i><font color="Green" title="E46">A72</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:3_2_1_7_2_1_1_2"/><b>then </b><i><font color="Green" title="E49">A75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20" title="BINTREE2:th.20">BINTREE2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">zb</font> = <font color="Maroon" title="c7">dD</font> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c18">n</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="bintree2.html#T10" title="BINTREE2:th.10">BINTREE2:10</a></i>;<br/>
<a NAME="E18:3_2_1_7_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">zb</font>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E16:3_2_1_7_2_1_1_2"><i><font color="Green" title="E49">A75</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">F</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c18">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="bintree2.html#T5" title="BINTREE2:th.5">BINTREE2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">F1</font> = <font color="Maroon" title="c21">F</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E21:3_2_1_7_2_1_1_2"/>
<font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E9:3_2_1_7_2_1_1_2"><i><font color="Green" title="E45">A71</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>;<br/>
<a NAME="E22:3_2_1_7_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/>
<a NAME="E23:3_2_1_7_2_1_1_2"/><b>then </b><i><font color="Green" title="E50">A76</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font>
 
<b>by </b><i><a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></i>;<br/>
<a NAME="E24:3_2_1_7_2_1_1_2"/>
<font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E25:3_2_1_7_2_1_1_2"/><b>then </b>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="vectmetr.html#E13:3_2_1_7_2_1_1_2"><i><font color="Green" title="E47">A73</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E26:3_2_1_7_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T15" title="NAT_2:th.15">NAT_2:15</a></i>;<br/>
<a NAME="E27:3_2_1_7_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1_7_2_1_1_2"><i><font color="Green" title="E50">A76</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E28:3_2_1_7_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20" title="BINTREE2:th.20">BINTREE2:20</a></i>;<br/>
<a NAME="E29:3_2_1_7_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">zb</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">FF</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c18">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="bintree2.html#T5" title="BINTREE2:th.5">BINTREE2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">FF1</font> = <font color="Maroon" title="c23">FF</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">a</font>, <font color="Maroon" title="c26">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E33:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E51">A77</font></i>: 
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E34:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E52">A78</font></i>: 
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E35:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E53">A79</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c22">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">a</font>,<font color="Maroon" title="c26">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">a'</font>, <font color="Maroon" title="c28">b'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E37:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E54">A80</font></i>: 
<font color="Maroon" title="c27">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E38:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E55">A81</font></i>: 
<font color="Maroon" title="c28">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E39:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E56">A82</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c24">FF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">a'</font>,<font color="Maroon" title="c28">b'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">a1</font> = <font color="Maroon" title="c25">a</font>, <font color="Maroon" title="c30">b1</font> = <font color="Maroon" title="c26">b</font>, <font color="Maroon" title="c31">a1'</font> = <font color="Maroon" title="c27">a'</font>, <font color="Maroon" title="c32">b1'</font> = <font color="Maroon" title="c28">b'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="vectmetr.html#E33:3_2_1_7_2_1_1_2"><i><font color="Green" title="E51">A77</font></i></a>, <a class="txt" href="vectmetr.html#E34:3_2_1_7_2_1_1_2"><i><font color="Green" title="E52">A78</font></i></a>, <a class="txt" href="vectmetr.html#E37:3_2_1_7_2_1_1_2"><i><font color="Green" title="E54">A80</font></i></a>, <a class="txt" href="vectmetr.html#E38:3_2_1_7_2_1_1_2"><i><font color="Green" title="E55">A81</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c33">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E42:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E57">A83</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">F1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">a1</font>,<font color="Maroon" title="c33">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E43:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E58">A84</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">F1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c33">d</font>,<font color="Maroon" title="c30">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E44:3_2_1_7_2_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c29">a1</font>,<font color="Maroon" title="c30">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c29">a1</font>,<font color="Maroon" title="c33">d</font></span>)</span>
 <b>and </b><br/><a NAME="E45:3_2_1_7_2_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c29">a1</font>,<font color="Maroon" title="c30">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c33">d</font>,<font color="Maroon" title="c30">b1</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="vectmetr.html#E35:3_2_1_7_2_1_1_2"><i><font color="Green" title="E53">A79</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c34">d'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E47:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E59">A85</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">FF1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c31">a1'</font>,<font color="Maroon" title="c34">d'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E48:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E60">A86</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">FF1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c34">d'</font>,<font color="Maroon" title="c32">b1'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E49:3_2_1_7_2_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">a1'</font>,<font color="Maroon" title="c32">b1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">a1'</font>,<font color="Maroon" title="c34">d'</font></span>)</span>
 <b>and </b><br/><a NAME="E50:3_2_1_7_2_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">a1'</font>,<font color="Maroon" title="c32">b1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c34">d'</font>,<font color="Maroon" title="c32">b1'</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="vectmetr.html#E39:3_2_1_7_2_1_1_2"><i><font color="Green" title="E56">A82</font></i></a></i>;<br/>
<a NAME="E51:3_2_1_7_2_1_1_2"/><i><font color="Green" title="E61">A87</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<i><font color="Green" title="E62">A88</font></i>: <font color="Maroon" title="c19">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="vectmetr.html#E51:3_2_1_7_2_1_1_2"><i><font color="Green" title="E61">A87</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7_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_7_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_7_2_1_1_2_2_1"/>
(  not <font color="Maroon" title="c19">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or <font color="Maroon" title="c19">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:3_2_1_7_2_1_1_2_2_1_1"/>
not <font color="Maroon" title="c19">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><a NAME="E2:3_2_1_7_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E63">A89</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="nat_2.html#T24" title="NAT_2:th.24">NAT_2:24</a></i>;<br/><a NAME="E3:3_2_1_7_2_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="vectmetr.html#E52:3_2_1_7_2_1_1_2"><i><font color="Green" title="E62">A88</font></i></a>, <a class="ref" href="nat_2.html#T20" title="NAT_2:th.20">NAT_2:20</a></i>;<br/><a NAME="E4:3_2_1_7_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 <b>by </b><i><a class="ref" href="nat_2.html#T23" title="NAT_2:th.23">NAT_2:23</a></i>;<br/><a NAME="E5:3_2_1_7_2_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T28" title="NAT_2:th.28">NAT_2:28</a></i>;<br/><a NAME="E6:3_2_1_7_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E64">A90</font></i>: 
<font color="Maroon" title="c33">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">d'</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E42:3_2_1_7_2_1_1_2"><i><font color="Green" title="E57">A83</font></i></a>, <a class="txt" href="vectmetr.html#E47:3_2_1_7_2_1_1_2"><i><font color="Green" title="E59">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">FF</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&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#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E9:3_2_1_7_2_1_1_2"><i><font color="Green" title="E45">A71</font></i></a>, <a class="ref" href="bintree2.html#T24" title="BINTREE2:th.24">BINTREE2:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">FF</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_2_2_1_1"><i><font color="Green" title="E63">A89</font></i></a>, <a class="ref" href="nat_d.html#T21" title="NAT_D:th.21">NAT_D:21</a></i>
<br/>.= 
<font color="Maroon" title="c33">d</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E48:3_2_1_7_2_1_1_2"><i><font color="Green" title="E60">A86</font></i></a>, <a class="txt" href="vectmetr.html#E6:3_2_1_7_2_1_1_2_2_1_1"><i><font color="Green" title="E64">A90</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">F</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E42:3_2_1_7_2_1_1_2"><i><font color="Green" title="E57">A83</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">F</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E52:3_2_1_7_2_1_1_2"><i><font color="Green" title="E62">A88</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_2_2_1_1"><i><font color="Green" title="E63">A89</font></i></a>, <a class="ref" href="nat_2.html#T20" title="NAT_2:th.20">NAT_2:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_2"><i><font color="Green" title="E41">A67</font></i></a>, <a class="txt" href="vectmetr.html#E8:3_2_1_7_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a>, <a class="ref" href="bintree2.html#T24" title="BINTREE2:th.24">BINTREE2:24</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:3_2_1_7_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c19">i</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><div class="add"><a NAME="E2:3_2_1_7_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E63">A91</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_2.html#T23" title="NAT_2:th.23">NAT_2:23</a></i>;<br/><b>then </b><i><font color="Green" title="E64">A92</font></i>: 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_2"><i><font color="Green" title="E41">A67</font></i></a>, <a class="ref" href="nat_2.html#T20" title="NAT_2:th.20">NAT_2:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="ref" href="nat_d.html#T21" title="NAT_D:th.21">NAT_D:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_2"><i><font color="Green" title="E41">A67</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
;<br/><a NAME="E4:3_2_1_7_2_1_1_2_2_1_2"/><i><font color="Green" title="E65">A93</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_2"><i><font color="Green" title="E41">A67</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1

;<br/><a NAME="E6:3_2_1_7_2_1_1_2_2_1_2"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
 <b>by </b><i><a class="txt" href="vectmetr.html#E3:3_2_1_7_2_1_1_2_2_1_2"><i><font color="Green" title="E64">A92</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_7_2_1_1_2_2_1_2"><i><font color="Green" title="E65">A93</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E7:3_2_1_7_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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_2.html#T20" title="NAT_2:th.20">NAT_2:20</a></i>;<br/><a NAME="E8:3_2_1_7_2_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/><a NAME="E9:3_2_1_7_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E66">A94</font></i>: 
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><a NAME="E10:3_2_1_7_2_1_1_2_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E11:3_2_1_7_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" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="vectmetr.html#E9:3_2_1_7_2_1_1_2_2_1_2"><i><font color="Green" title="E66">A94</font></i></a>, <a class="ref" href="nat_2.html#T17" title="NAT_2:th.17">NAT_2:17</a></i>;<br/><a NAME="E12:3_2_1_7_2_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E13:3_2_1_7_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E67">A95</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 ;<br/><a NAME="E14:3_2_1_7_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E68">A96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1_7_2_1_1_2"><i><font color="Green" title="E50">A76</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><i><font color="Green" title="E69">A97</font></i>: <font color="Maroon" title="c27">a'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E39:3_2_1_7_2_1_1_2"><i><font color="Green" title="E56">A82</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_7_2_1_1_2"><i><font color="Green" title="E40">A66</font></i></a>, <a class="txt" href="vectmetr.html#E14:3_2_1_7_2_1_1_2"><i><font color="Green" title="E48">A74</font></i></a>, <a class="txt" href="vectmetr.html#E13:3_2_1_7_2_1_1_2_2_1_2"><i><font color="Green" title="E67">A95</font></i></a>, <a class="txt" href="vectmetr.html#E14:3_2_1_7_2_1_1_2_2_1_2"><i><font color="Green" title="E68">A96</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c26">b</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E35:3_2_1_7_2_1_1_2"><i><font color="Green" title="E53">A79</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">FF</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&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#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E9:3_2_1_7_2_1_1_2"><i><font color="Green" title="E45">A71</font></i></a>, <a class="ref" href="bintree2.html#T24" title="BINTREE2:th.24">BINTREE2:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">FF</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_2_2_1_2"><i><font color="Green" title="E63">A91</font></i></a>, <a class="ref" href="nat_d.html#T21" title="NAT_D:th.21">NAT_D:21</a></i>
<br/>.= 
<font color="Maroon" title="c31">a1'</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E47:3_2_1_7_2_1_1_2"><i><font color="Green" title="E59">A85</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">F</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E43:3_2_1_7_2_1_1_2"><i><font color="Green" title="E58">A84</font></i></a>, <a class="txt" href="vectmetr.html#E15:3_2_1_7_2_1_1_2_2_1_2"><i><font color="Green" title="E69">A97</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_2_1_1_2"><i><font color="Green" title="E41">A67</font></i></a>, <a class="txt" href="vectmetr.html#E8:3_2_1_7_2_1_1_2"><i><font color="Green" title="E44">A70</font></i></a>, <a class="txt" href="vectmetr.html#E3:3_2_1_7_2_1_1_2_2_1_2"><i><font color="Green" title="E64">A92</font></i></a>, <a class="ref" href="bintree2.html#T24" title="BINTREE2:th.24">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="E54:3_2_1_7_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E8:3_2_1_7_2_1_1"/><i><font color="Green" title="E40">A98</font></i>: 
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="vectmetr.html#E6:3_2_1_7_2_1_1"><i><font color="Green" title="E38">A59</font></i></a>, <a class="txt" href="vectmetr.html#E7:3_2_1_7_2_1_1"><i><font color="Green" title="E39">A65</font></i></a>);<br/></i><a NAME="E9:3_2_1_7_2_1_1"/><i><font color="Green" title="E41">A99</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E10:3_2_1_7_2_1_1"/><b>then </b><i><font color="Green" title="E42">A100</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_7_2_1_1"><i><font color="Green" title="E36">A57</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:3_2_1_7_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">h</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c14">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_7_2_1_1"><i><font color="Green" title="E36">A57</font></i></a>, <a class="txt" href="vectmetr.html#E9:3_2_1_7_2_1_1"><i><font color="Green" title="E41">A99</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E24:3_2_1"><i><font color="Green" title="E14">A19</font></i></a>, <a class="txt" href="vectmetr.html#E10:3_2_1_7_2_1_1"><i><font color="Green" title="E42">A100</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E20:3_2_1"><i><font color="Green" title="E11">A16</font></i></a>, <a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_7_2_1_1"><i><font color="Green" title="E36">A57</font></i></a>, <a class="txt" href="vectmetr.html#E9:3_2_1_7_2_1_1"><i><font color="Green" title="E41">A99</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">FSL</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E21:3_2_1"><i><font color="Green" title="E12">A17</font></i></a>, <a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E10:3_2_1_7_2_1_1"><i><font color="Green" title="E42">A100</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E3:3_2_1_7"><i><font color="Green" title="E30">A51</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_7_2_1_1"><i><font color="Green" title="E36">A57</font></i></a>, <a class="txt" href="vectmetr.html#E9:3_2_1_7_2_1_1"><i><font color="Green" title="E41">A99</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_7"><i><font color="Green" title="E28">A49</font></i></a>, <a class="txt" href="vectmetr.html#E5:3_2_1_7_2_1_1"><i><font color="Green" title="E37">A58</font></i></a>, <a class="txt" href="vectmetr.html#E8:3_2_1_7_2_1_1"><i><font color="Green" title="E40">A98</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_2_1_2"><b>suppose </b></a><a NAME="E1:3_2_1_7_2_1_2"/><i><font color="Green" title="E36">A101</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c17">ze</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E31:3_2_1"><i><font color="Green" title="E20">A41</font></i></a>, <a class="txt" href="vectmetr.html#E42:3_2_1"><i><font color="Green" title="E26">A47</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E1:3_2_1_7_2_1_2"><i><font color="Green" title="E36">A101</font></i></a>, <a class="ref" href="bintree2.html#T16" title="BINTREE2:th.16">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_7"/><i><font color="Green" title="E36">A102</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_7"><i><font color="Green" title="E28">A49</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[ non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">j</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> $1 holds <br/>for <font color="Olive" title="b2">DF1</font>, <font color="Olive" title="b3">DF2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">DF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> $1,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Olive" title="b3">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> $1,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b2">DF1</font>,<font color="Olive" title="b3">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> $1</span>)</span>;<br/>
<a NAME="E14:3_2_1_7"/><i><font color="Green" title="E37">A103</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">j</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:3_2_1_7_3"/><i><font color="Green" title="E38">A104</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/>
<a NAME="E2:3_2_1_7_3"/><i><font color="Green" title="E39">A105</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>assume </b><a NAME="E3:3_2_1_7_3"/><i><font color="Green" title="E40">A106</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1
 ;<br/>

<a NAME="E4:3_2_1_7_3"/><b>then </b><i><font color="Green" title="E41">A107</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_7_3"><i><font color="Green" title="E38">A104</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_1_7_3"><i><font color="Green" title="E39">A105</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>let </b><font color="Maroon" title="c19">DF1</font>, <font color="Maroon" title="c20">DF2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E5:3_2_1_7_3"/><i><font color="Green" title="E42">A108</font></i>: 
<font color="Maroon" title="c19">DF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> 1,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>and </b><br/><a NAME="E6:3_2_1_7_3"/><i><font color="Green" title="E43">A109</font></i>: 
<font color="Maroon" title="c20">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> 1,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c21">pusty</font> =  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E8:3_2_1_7_3"/>
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E9:3_2_1_7_3"/><b>then </b>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E3:3_2_1_7_3"><i><font color="Green" title="E40">A106</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:3_2_1_7_3"/><b>then </b>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> 1,<font color="Maroon" title="c7">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20" title="BINTREE2:th.20">BINTREE2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">FSL1j</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> 1,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_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_7_3_1_1"/>
( <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="vectmetr.html#E4:3_2_1_7_3"><i><font color="Green" title="E41">A107</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_3_1_1_1"><b>suppose </b></a><a NAME="E1:3_2_1_7_3_1_1_1"/><i><font color="Green" title="E44">A110</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><i><font color="Green" title="E45">A111</font></i>: <font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c22">FSL1j</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_7_3_1_1_1"><i><font color="Green" title="E44">A110</font></i></a>, <a class="ref" href="bintree2.html#T22" title="BINTREE2:th.22">BINTREE2:22</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">DF1</font>,<font color="Maroon" title="c20">DF2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="vectmetr.html#E5:3_2_1_7_3"><i><font color="Green" title="E42">A108</font></i></a>, <a class="txt" href="vectmetr.html#E6:3_2_1_7_3"><i><font color="Green" title="E43">A109</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>
;<br/><b>consider </b><font color="Maroon" title="c23">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:3_2_1_7_3_1_1_1"/><i><font color="Green" title="E46">A112</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c23">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E5:3_2_1_7_3_1_1_1"/>
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E6:3_2_1_7_3_1_1_1"/><i><font color="Green" title="E47">A113</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c23">b</font></span>)</span>
 <b>and </b><br/><a NAME="E7:3_2_1_7_3_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E6:3"><i><font color="Green" title="E7">A12</font></i></a>, <a class="txt" href="vectmetr.html#E7:3"><i><font color="Green" title="E8">A13</font></i></a></i>;<br/><a NAME="E8:3_2_1_7_3_1_1_1"/>
( <font color="Maroon" title="c19">DF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c20">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b</font> )
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_3_1_1_1"><i><font color="Green" title="E45">A111</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_7_3_1_1_1"><i><font color="Green" title="E46">A112</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E9:3_2_1_7_3_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c19">DF1</font>,<font color="Maroon" title="c20">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="vectmetr.html#E6:3_2_1_7_3_1_1_1"><i><font color="Green" title="E47">A113</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_3_1_1_2"><b>suppose </b></a><a NAME="E1:3_2_1_7_3_1_1_2"/><i><font color="Green" title="E44">A114</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><i><font color="Green" title="E45">A115</font></i>: <font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c22">FSL1j</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_7_3_1_1_2"><i><font color="Green" title="E44">A114</font></i></a>, <a class="ref" href="bintree2.html#T23" title="BINTREE2:th.23">BINTREE2:23</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">DF1</font>,<font color="Maroon" title="c20">DF2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="vectmetr.html#E5:3_2_1_7_3"><i><font color="Green" title="E42">A108</font></i></a>, <a class="txt" href="vectmetr.html#E6:3_2_1_7_3"><i><font color="Green" title="E43">A109</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>
;<br/><b>consider </b><font color="Maroon" title="c23">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:3_2_1_7_3_1_1_2"/>
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c23">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E5:3_2_1_7_3_1_1_2"/><i><font color="Green" title="E46">A116</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pusty</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E6:3_2_1_7_3_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c23">b</font></span>)</span>
 <b>and </b><br/><a NAME="E7:3_2_1_7_3_1_1_2"/><i><font color="Green" title="E47">A117</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E6:3"><i><font color="Green" title="E7">A12</font></i></a>, <a class="txt" href="vectmetr.html#E7:3"><i><font color="Green" title="E8">A13</font></i></a></i>;<br/><a NAME="E8:3_2_1_7_3_1_1_2"/>
( <font color="Maroon" title="c19">DF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b</font> &amp; <font color="Maroon" title="c20">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_3_1_1_2"><i><font color="Green" title="E45">A115</font></i></a>, <a class="txt" href="vectmetr.html#E5:3_2_1_7_3_1_1_2"><i><font color="Green" title="E46">A116</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E9:3_2_1_7_3_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c19">DF1</font>,<font color="Maroon" title="c20">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3_2_1_7_3_1_1_2"><i><font color="Green" title="E47">A117</font></i></a></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_7_3"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c19">DF1</font>,<font color="Maroon" title="c20">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:3_2_1_7"/><i><font color="Green" title="E38">A118</font></i>: 
for <font color="Olive" title="b1">l</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">l</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">l</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3_2_1_7_4"/><i><font color="Green" title="E39">A119</font></i>: 
for <font color="Olive" title="b1">j</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font> holds <br/>for <font color="Olive" title="b2">DF1</font>, <font color="Olive" title="b3">DF2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">DF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">l</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Olive" title="b3">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">l</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b2">DF1</font>,<font color="Olive" title="b3">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c19">j</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:3_2_1_7_4"/><i><font color="Green" title="E40">A120</font></i>: 
<font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c20">DF1</font>, <font color="Maroon" title="c21">DF2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:3_2_1_7_4"/><i><font color="Green" title="E41">A121</font></i>: 
<font color="Maroon" title="c20">DF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>and </b><br/><a NAME="E4:3_2_1_7_4"/><i><font color="Green" title="E42">A122</font></i>: 
<font color="Maroon" title="c21">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ;<br/>

<a NAME="E5:3_2_1_7_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_2_1_7_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:3_2_1_7_4_1"/><b>then </b>
<font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T14" title="NAT_2:th.14">NAT_2:14</a></i>;<br/>
<a NAME="E3:3_2_1_7_4_1"/><b>then </b>
<font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E4:3_2_1_7_4_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c22">j1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E7:3_2_1_7_4"/><i><font color="Green" title="E43">A123</font></i>: 
<font color="Maroon" title="c22">j1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E8:3_2_1_7_4"/>
<font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_4"><i><font color="Green" title="E40">A120</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>;<br/>
<a NAME="E9:3_2_1_7_4"/><b>then </b>
<font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
 
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/>
<a NAME="E10:3_2_1_7_4"/><b>then </b><i><font color="Green" title="E44">A124</font></i>: 
<font color="Maroon" title="c22">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font>
 
<b>by </b><i><a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></i>;<br/>
<a NAME="E11:3_2_1_7_4"/><b>then </b>
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E7:3_2_1_7_4"><i><font color="Green" title="E43">A123</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:3_2_1_7_4"/><b>then </b><i><font color="Green" title="E45">A125</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">l</font>,<font color="Maroon" title="c7">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20" title="BINTREE2:th.20">BINTREE2:20</a></i>;<br/>
<a NAME="E13:3_2_1_7_4"/>
<font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E14:3_2_1_7_4"/><b>then </b>
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_4"><i><font color="Green" title="E40">A120</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:3_2_1_7_4"/><b>then </b><i><font color="Green" title="E46">A126</font></i>: 
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bintree2.html#T20" title="BINTREE2:th.20">BINTREE2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">dDll1</font> = <font color="Maroon" title="c7">dD</font> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="bintree2.html#T10" title="BINTREE2:th.10">BINTREE2:10</a></i>;<br/>
<a NAME="E17:3_2_1_7_4"/>
<span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">dDll1</font>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E15:3_2_1_7_4"><i><font color="Green" title="E46">A126</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">FSLl1</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="bintree2.html#T5" title="BINTREE2:th.5">BINTREE2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">Fj</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="vectmetr.html#E15:3_2_1_7_4"><i><font color="Green" title="E46">A126</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E20:3_2_1_7_4"/><i><font color="Green" title="E47">A127</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c25">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">DF1</font>,<font color="Maroon" title="c21">DF2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E3:3_2_1_7_4"><i><font color="Green" title="E41">A121</font></i></a>, <a class="txt" href="vectmetr.html#E4:3_2_1_7_4"><i><font color="Green" title="E42">A122</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">FSLl</font> being   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c18">l</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> , <font color="Maroon" title="c27">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E22:3_2_1_7_4"/><i><font color="Green" title="E48">A128</font></i>: 
<font color="Maroon" title="c24">FSLl1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">FSLl</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c27">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137" title="FINSEQ_2:th.137">FINSEQ_2:137</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">NFSLl</font> = <font color="Maroon" title="c26">FSLl</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">x1</font>, <font color="Maroon" title="c30">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E25:3_2_1_7_4"/><i><font color="Green" title="E49">A129</font></i>: 
<font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E26:3_2_1_7_4"/><i><font color="Green" title="E50">A130</font></i>: 
<font color="Maroon" title="c30">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E27:3_2_1_7_4"/><i><font color="Green" title="E51">A131</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c28">NFSLl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<font color="Maroon" title="c30">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">x1</font> = <font color="Maroon" title="c29">x1</font>, <font color="Maroon" title="c32">y1</font> = <font color="Maroon" title="c30">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="vectmetr.html#E25:3_2_1_7_4"><i><font color="Green" title="E49">A129</font></i></a>, <a class="txt" href="vectmetr.html#E26:3_2_1_7_4"><i><font color="Green" title="E50">A130</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c33">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E30:3_2_1_7_4"/><i><font color="Green" title="E52">A132</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">NFSLl</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c31">x1</font>,<font color="Maroon" title="c33">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E31:3_2_1_7_4"/><i><font color="Green" title="E53">A133</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">NFSLl</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c33">b</font>,<font color="Maroon" title="c32">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E32:3_2_1_7_4"/><i><font color="Green" title="E54">A134</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">x1</font>,<font color="Maroon" title="c32">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">x1</font>,<font color="Maroon" title="c33">b</font></span>)</span>
 <b>and </b><br/><a NAME="E33:3_2_1_7_4"/><i><font color="Green" title="E55">A135</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">x1</font>,<font color="Maroon" title="c32">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c33">b</font>,<font color="Maroon" title="c32">y1</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:3"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="vectmetr.html#E27:3_2_1_7_4"><i><font color="Green" title="E51">A131</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">dDll</font> = <font color="Maroon" title="c7">dD</font> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c18">l</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="vectmetr.html#E5:3"><i><font color="Green" title="E6">A11</font></i></a>, <a class="ref" href="bintree2.html#T10" title="BINTREE2:th.10">BINTREE2:10</a></i>;<br/>
<a NAME="E35:3_2_1_7_4"/>
<span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">l</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">dDll</font>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E12:3_2_1_7_4"><i><font color="Green" title="E45">A125</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c35">FSLlprim</font> = <span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">l</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c18">l</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="bintree2.html#T5" title="BINTREE2:th.5">BINTREE2:5</a></i>;<br/>
<a NAME="E37:3_2_1_7_4"/>
<span class="p1">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">FSLlprim</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_1_7_4"><i><font color="Green" title="E40">A120</font></i></a>, <a class="ref" href="bintree2.html#T24" title="BINTREE2:th.24">BINTREE2:24</a></i>;<br/>
<a NAME="E38:3_2_1_7_4"/><b>then </b><i><font color="Green" title="E56">A136</font></i>: 
( <font color="Maroon" title="c26">FSLl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">FSLlprim</font> &amp; <font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 )
 
<b>by </b><i><a class="txt" href="vectmetr.html#E22:3_2_1_7_4"><i><font color="Green" title="E48">A128</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E39:3_2_1_7_4"/><b>then </b><i><font color="Green" title="E57">A137</font></i>: 
( <font color="Maroon" title="c31">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">l</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j1</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c32">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bintree2.html#K2" title="BINTREE2:func.2">FinSeqLevel</a> <font color="Maroon" title="c18">l</font>,<font color="Maroon" title="c7">dD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 
<b>by </b><i><a class="txt" href="vectmetr.html#E27:3_2_1_7_4"><i><font color="Green" title="E51">A131</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_7_4_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_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_7_4_2_1"/>
( <font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="vectmetr.html#E38:3_2_1_7_4"><i><font color="Green" title="E56">A136</font></i></a>, <a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_4_2_1_1"><b>suppose </b></a><a NAME="E1:3_2_1_7_4_2_1_1"/>
<font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:3_2_1_7_4_2_1_1"/><b>then </b>
( <font color="Maroon" title="c20">DF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x1</font> &amp; <font color="Maroon" title="c21">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">b</font> )
 <b>by </b><i><a class="txt" href="vectmetr.html#E20:3_2_1_7_4"><i><font color="Green" title="E47">A127</font></i></a>, <a class="txt" href="vectmetr.html#E22:3_2_1_7_4"><i><font color="Green" title="E48">A128</font></i></a>, <a class="txt" href="vectmetr.html#E30:3_2_1_7_4"><i><font color="Green" title="E52">A132</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E3:3_2_1_7_4_2_1_1"/><b>then </b>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">DF1</font>,<font color="Maroon" title="c21">DF2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_7_4"><i><font color="Green" title="E39">A119</font></i></a>, <a class="txt" href="vectmetr.html#E10:3_2_1_7_4"><i><font color="Green" title="E44">A124</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1_7_4"><i><font color="Green" title="E54">A134</font></i></a>, <a class="txt" href="vectmetr.html#E39:3_2_1_7_4"><i><font color="Green" title="E57">A137</font></i></a></i>;<br/><b>hence </b> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">DF1</font>,<font color="Maroon" title="c21">DF2</font> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T79" title="XCMPLX_1:th.79">XCMPLX_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_1_7_4_2_1_2"><b>suppose </b></a><a NAME="E1:3_2_1_7_4_2_1_2"/>
<font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:3_2_1_7_4_2_1_2"/><b>then </b>
( <font color="Maroon" title="c20">DF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">b</font> &amp; <font color="Maroon" title="c21">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">y1</font> )
 <b>by </b><i><a class="txt" href="vectmetr.html#E20:3_2_1_7_4"><i><font color="Green" title="E47">A127</font></i></a>, <a class="txt" href="vectmetr.html#E22:3_2_1_7_4"><i><font color="Green" title="E48">A128</font></i></a>, <a class="txt" href="vectmetr.html#E31:3_2_1_7_4"><i><font color="Green" title="E53">A133</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E3:3_2_1_7_4_2_1_2"/><b>then </b>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">DF1</font>,<font color="Maroon" title="c21">DF2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_1_7_4"><i><font color="Green" title="E39">A119</font></i></a>, <a class="txt" href="vectmetr.html#E10:3_2_1_7_4"><i><font color="Green" title="E44">A124</font></i></a>, <a class="txt" href="vectmetr.html#E33:3_2_1_7_4"><i><font color="Green" title="E55">A135</font></i></a>, <a class="txt" href="vectmetr.html#E39:3_2_1_7_4"><i><font color="Green" title="E57">A137</font></i></a></i>;<br/><b>hence </b> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">DF1</font>,<font color="Maroon" title="c21">DF2</font> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T79" title="XCMPLX_1:th.79">XCMPLX_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c18">l</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">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="E41:3_2_1_7_4"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">DF1</font>,<font color="Maroon" title="c21">DF2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:3_2_1_7"/>
for <font color="Olive" title="b1">l</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">l</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="vectmetr.html#E14:3_2_1_7"><i><font color="Green" title="E37">A103</font></i></a>, <a class="txt" href="vectmetr.html#E15:3_2_1_7"><i><font color="Green" title="E38">A118</font></i></a>);<br/></i>
<b>hence </b><a NAME="E17:3_2_1_7"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_1_7"><i><font color="Green" title="E29">A50</font></i></a>, <a class="txt" href="vectmetr.html#E6:3_2_1_7"><i><font color="Green" title="E32">A53</font></i></a>, <a class="txt" href="vectmetr.html#E12:3_2_1_7"><i><font color="Green" title="E35">A56</font></i></a>, <a class="txt" href="vectmetr.html#E13:3_2_1_7"><i><font color="Green" title="E36">A102</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E44:3_2_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E17:3_2_1"><i><font color="Green" title="E10">A15</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c16">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E45:3_2_1"/><i><font color="Green" title="E28">A138</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>and </b><br/><a NAME="E46:3_2_1"/><i><font color="Green" title="E29">A139</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> holds <br/><font color="Maroon" title="c16">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E47:3_2_1"/><i><font color="Green" title="E30">A140</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E48:3_2_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c16">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_2_1_8"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c16">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:3_2_1_8_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c16">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_2_1_8_1"/><i><font color="Green" title="E31">A141</font></i>: 
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">F</font>
 <b>and </b><br/><a NAME="E4:3_2_1_8_1"/><i><font color="Green" title="E32">A142</font></i>: 
<font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:3_2_1_8_1"/>
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E3:3_2_1_8_1"><i><font color="Green" title="E31">A141</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:3_2_1_8_1"/><b>then </b>
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">c1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:3_2_1_8_1"/><i><font color="Green" title="E33">A143</font></i>: 
<font color="Maroon" title="c19">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">c</font>
 <b>and </b><br/><a NAME="E9:3_2_1_8_1"/><i><font color="Green" title="E34">A144</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">c1</font>
 <b>and </b><br/><a NAME="E10:3_2_1_8_1"/><i><font color="Green" title="E35">A145</font></i>: 
<font color="Maroon" title="c19">c1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font>
 ;<br/>
<font color="Maroon" title="c17">a</font> = 
<font color="Maroon" title="c16">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">c1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:3_2_1_8_1"><i><font color="Green" title="E32">A142</font></i></a>, <a class="txt" href="vectmetr.html#E8:3_2_1_8_1"><i><font color="Green" title="E33">A143</font></i></a>, <a class="txt" href="vectmetr.html#E9:3_2_1_8_1"><i><font color="Green" title="E34">A144</font></i></a>, <a class="txt" href="vectmetr.html#E10:3_2_1_8_1"><i><font color="Green" title="E35">A145</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">c1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">c1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E46:3_2_1"><i><font color="Green" title="E29">A139</font></i></a>, <a class="txt" href="vectmetr.html#E9:3_2_1_8_1"><i><font color="Green" title="E34">A144</font></i></a>, <a class="txt" href="vectmetr.html#E10:3_2_1_8_1"><i><font color="Green" title="E35">A145</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E43:3_2_1"><i><font color="Green" title="E27">A48</font></i></a>, <a class="txt" href="vectmetr.html#E45:3_2_1"><i><font color="Green" title="E28">A138</font></i></a>, <a class="txt" href="vectmetr.html#E9:3_2_1_8_1"><i><font color="Green" title="E34">A144</font></i></a>, <a class="txt" href="vectmetr.html#E10:3_2_1_8_1"><i><font color="Green" title="E35">A145</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E12:3_2_1_8_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c17">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:3_2_1_8"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E3:3_2_1_8"/><b>then </b><i><font color="Green" title="E31">A146</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:3_2_1_8"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E33:3_2_1"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="vectmetr.html#E45:3_2_1"><i><font color="Green" title="E28">A138</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:3_2_1_8"/><b>then </b><i><font color="Green" title="E32">A147</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<font color="Maroon" title="c17">a</font> = 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E33:3_2_1"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="vectmetr.html#E43:3_2_1"><i><font color="Green" title="E27">A48</font></i></a>, <a class="txt" href="vectmetr.html#E3:3_2_1_8"><i><font color="Green" title="E31">A146</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c16">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E33:3_2_1"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="vectmetr.html#E45:3_2_1"><i><font color="Green" title="E28">A138</font></i></a>, <a class="txt" href="vectmetr.html#E46:3_2_1"><i><font color="Green" title="E29">A139</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c16">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E33:3_2_1"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="vectmetr.html#E45:3_2_1"><i><font color="Green" title="E28">A138</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/>
<b>hence </b><a NAME="E7:3_2_1_8"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c16">F</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E5:3_2_1_8"><i><font color="Green" title="E32">A147</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E49:3_2_1"/><b>then </b>
<font color="Maroon" title="c16">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E47:3_2_1"><i><font color="Green" title="E30">A140</font></i></a>, <a class="ref" href="funcop_1.html#T15" title="FUNCOP_1:th.15">FUNCOP_1:15</a></i>;<br/><b>hence </b> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c16">F</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c10">N</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E23:3_2_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="vectmetr.html#E32:3_2_1"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="vectmetr.html#E45:3_2_1"><i><font color="Green" title="E28">A138</font></i></a>, <a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>
<br/>.= 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">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" title="E9">A148</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">f</font> = <span class="p1"><a href="rlvect_1.html#K6" title="RLVECT_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="rlvect_1.html#K6" title="RLVECT_1:func.6">*&gt;</a></span>;<br/><b>thus </b><a NAME="E2:3_2_2"/><i><font color="Green" title="E10">A149</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><a NAME="E3:3_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E4:3_2_2"/>
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><i><font color="Green" title="E11">A150</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
1

;<br/><b>thus </b><a NAME="E6:3_2_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">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" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:3_2_2_2"/><i><font color="Green" title="E12">A151</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <b>and </b><br/><a NAME="E2:3_2_2_2"/><i><font color="Green" title="E13">A152</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ;<br/>

<a NAME="E3:3_2_2_2"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="txt" href="vectmetr.html#E5:3_2_2"><i><font color="Green" title="E11">A150</font></i></a>, <a class="txt" href="vectmetr.html#E1:3_2_2_2"><i><font color="Green" title="E12">A151</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_2_2"><i><font color="Green" title="E13">A152</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:3_2_2_2"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:3_2_2_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:3_2_2"><i><font color="Green" title="E9">A148</font></i></a>, <a class="txt" href="vectmetr.html#E2:3_2_2"><i><font color="Green" title="E10">A149</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c9">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E7:3_2_2"/><i><font color="Green" title="E12">A153</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>and </b><br/><a NAME="E8:3_2_2"/><i><font color="Green" title="E13">A154</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F</font> holds <br/><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E5:3_2_2"><i><font color="Green" title="E11">A150</font></i></a>, <a class="txt" href="vectmetr.html#E7:3_2_2"><i><font color="Green" title="E12">A153</font></i></a>, <a class="txt" href="vectmetr.html#E8:3_2_2"><i><font color="Green" title="E13">A154</font></i></a></i>
<br/>.= 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:3_2_2"><i><font color="Green" title="E10">A149</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>
;<br/><a NAME="E10:3_2_2"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E5:3_2_2"><i><font color="Green" title="E11">A150</font></i></a>, <a class="txt" href="vectmetr.html#E7:3_2_2"><i><font color="Green" title="E12">A153</font></i></a>, <a class="ref" href="finseq_5.html#T15" title="FINSEQ_5:th.15">FINSEQ_5:15</a></i>;<br/><b>hence </b><a NAME="E11:3_2_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">F</font>
 <b>by </b><i><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
