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

<b>let </b><font color="Maroon" title="c1">B1</font>, <font color="Maroon" title="c2">B2</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">B1</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c2">B2</font> holds <br/> <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Olive" title="b1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">B1</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c2">B2</font> implies  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:67"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c1">B1</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c2">B2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c4">B10</font> = <font color="Maroon" title="c1">B1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">B20</font> = <font color="Maroon" title="c2">B2</font>;<br/>
<b>set </b><font color="Maroon" title="c6">X0</font> = <font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">Y0</font> = <font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>
<a NAME="E2:67"/><i><font color="Green" title="E44">G55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="membered.html#T6" title="MEMBERED:th.6">MEMBERED:6</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="afinsq_2.html#T36" target="_self" title="AFINSQ_2:th.36">AE400</a>, <a class="txt" href="#E1:67"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="afinsq_2.html#T34" target="_self" title="AFINSQ_2:th.34">AG110</a></i>
;<br/>
<a NAME="E4:67"/><b>then </b>
<font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E5:67"/><b>then </b><i><font color="Green" title="E45">A92</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">f2</font> =  <a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_2"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></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>  or  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1"><b>suppose </b></a><a NAME="E1:67_2_1"/><i><font color="Green" title="E46">C1</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span><br/><div class="add"><b>set </b><font color="Maroon" title="c9">f3</font> =  <a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font>;<br/><a NAME="E2:67_2_1"/><i><font color="Green" title="E47">F3</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_2_1_1"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></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>  or  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1"><b>suppose </b></a><a NAME="E1:67_2_1_1_1"/><i><font color="Green" title="E48">D1</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span><br/><div class="add"><b>set </b><font color="Maroon" title="c10">f4</font> =  <a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font>;<br/><a NAME="E2:67_2_1_1_1"/><i><font color="Green" title="E49">F4</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><a NAME="E3:67_2_1_1_1"/><i><font color="Green" title="E50">A1b</font></i>: 
<font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:67_2_1_1_1"><i><font color="Green" title="E48">D1</font></i></a>, <a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_2_1_1_1_1"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></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>  or  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_2_1_1_1_1_1"/><i><font color="Green" title="E51">E1</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span><br/><div class="add"><b>consider </b><font color="Maroon" title="c11">g1</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:67_2_1_1_1_1_1"/><i><font color="Green" title="E52">E2</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font> &amp; <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> holds <br/><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp;  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="afinsq_2.html#D4" target="_self" title="AFINSQ_2:def.4">AC300</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">g2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E5:67_2_1_1_1_1_1"/><i><font color="Green" title="E53">E3</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">g2</font> &amp; <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> holds <br/><font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp;  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="afinsq_2.html#D4" target="_self" title="AFINSQ_2:def.4">AC300</a></i>;<br/><b>set </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c14">g3</font> = <font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>;<br/><a NAME="E6:67_2_1_1_1_1_1"/><i><font color="Green" title="E54">E4</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">g2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="afinsq_2.html#T6" target="_self" title="AFINSQ_2:th.6">AC500</a></i>;<br/><a NAME="E7:67_2_1_1_1_1_1"/><i><font color="Green" title="E55">E7</font></i>: 
<font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T35" target="_self" title="AFINSQ_2:th.35">AE101</a>, <a class="txt" href="#E1:67"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="xboole_1.html#T76" title="XBOOLE_1:th.76">XBOOLE_1:76</a></i>;<br/><b>set </b><font color="Maroon" title="c15">g4</font> = <font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>;<br/><i><font color="Green" title="E56">E8b</font></i>: <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="txt" href="#E5:67_2_1_1_1_1_1"><i><font color="Green" title="E53">E3</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1"><i><font color="Green" title="E54">E4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:67_2_1_1_1_1_1"><i><font color="Green" title="E55">E7</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>
;<br/><a NAME="E9:67_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E57">E8</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></i>;<br/><a NAME="E10:67_2_1_1_1_1_1"/><i><font color="Green" title="E58">F4h</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><a NAME="E11:67_2_1_1_1_1_1"/><i><font color="Green" title="E59">E44</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font>
 <b>by </b><i><a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="txt" href="#E1:67_2_1_1_1"><i><font color="Green" title="E48">D1</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E12:67_2_1_1_1_1_1"/><i><font color="Green" title="E60">E77n</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="afinsq_2.html#T45" target="_self" title="AFINSQ_2:th.45">AE220</a>, <a class="txt" href="#E3:67_2_1_1_1"><i><font color="Green" title="E50">A1b</font></i></a>, <a class="ref" href="afinsq_2.html#T36" target="_self" title="AFINSQ_2:th.36">AE400</a>, <a class="txt" href="#E1:67"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/><i><font color="Green" title="E61">E9</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E11:67_2_1_1_1_1_1"><i><font color="Green" title="E59">E44</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E1:67_2_1_1_1"><i><font color="Green" title="E48">D1</font></i></a>, <a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E12:67_2_1_1_1_1_1"><i><font color="Green" title="E60">E77n</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E1:67_2_1"><i><font color="Green" title="E46">C1</font></i></a></i>
;<br/><a NAME="E14:67_2_1_1_1_1_1"/><i><font color="Green" title="E62">E89</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_1_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:67_2_1_1_1_1_1_3"/><i><font color="Green" title="E63">F1</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_2_1_1_1_1_1_3_1"/>
( <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font> or <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:67_2_1_1_1_1_1_3_1_1"/><i><font color="Green" title="E64">G1</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:67_2_1_1_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E3:67_2_1_1_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E65">G2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E4:67_2_1_1_1_1_1_3_1_1"/>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E5:67_2_1_1_1_1_1_3_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>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font>
 <b>by </b><i><a class="txt" href="#E1:67_2_1_1_1_1_1_3_1_1"><i><font color="Green" title="E64">G1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:67_2_1_1_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">g1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E7:67_2_1_1_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E66">G4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E8:67_2_1_1_1_1_1_3_1_1"/>
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th44</a>, <a class="txt" href="#E1:67_2_1_1_1_1_1_3_1_1"><i><font color="Green" title="E64">G1</font></i></a>, <a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="txt" href="#E2:67_2_1"><i><font color="Green" title="E47">F3</font></i></a></i>;<br/><a NAME="E9:67_2_1_1_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E67">G20</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T40" target="_self" title="AFINSQ_2:th.40">AE221</a>, <a class="txt" href="#E2:67"><i><font color="Green" title="E44">G55</font></i></a>, <a class="ref" href="afinsq_2.html#T36" target="_self" title="AFINSQ_2:th.36">AE400</a>, <a class="txt" href="#E1:67"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/><a NAME="E10:67_2_1_1_1_1_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="txt" href="#E1:67_2_1_1_1_1_1_3_1_1"><i><font color="Green" title="E64">G1</font></i></a></i>;<br/><a NAME="E11:67_2_1_1_1_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E9:67_2_1_1_1_1_1_3_1_1"><i><font color="Green" title="E67">G20</font></i></a>, <a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E1:67_2_1_1_1_1_1_3"><i><font color="Green" title="E63">F1</font></i></a></i>;<br/><b>hence </b><a NAME="E12:67_2_1_1_1_1_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:67_2_1_1_1_1_1_3_1_1"><i><font color="Green" title="E65">G2</font></i></a>, <a class="txt" href="#E7:67_2_1_1_1_1_1_3_1_1"><i><font color="Green" title="E66">G4</font></i></a>, <a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="txt" href="#E1:67_2_1_1_1_1_1_3_1_1"><i><font color="Green" title="E64">G1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:67_2_1_1_1_1_1_3_1_2"/><i><font color="Green" title="E64">G1</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><b>set </b><font color="Maroon" title="c17">X0</font> = <font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c18">Y0</font> = <font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1_1_1_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_2_1_1_1_1_1_3_1_2_1"/>
( <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font> or <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font> )
 </span><b>by </b><i><a class="txt" href="#E1:67_2_1_1_1_1_1_3_1_2"><i><font color="Green" title="E64">G1</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1_1_1_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:67_2_1_1_1_1_1_3_1_2_1_1"/>
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:67_2_1_1_1_1_1_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font>
 <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="E3:67_2_1_1_1_1_1_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">i1</font> = <font color="Maroon" title="c16">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><i><font color="Green" title="E65">G67</font></i>: <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>
;<br/><a NAME="E6:67_2_1_1_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E66">G3dn</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:67_2_1_1_1_1_1_3"><i><font color="Green" title="E63">F1</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:67_2_1_1_1_1_1_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:67_2_1_1_1_1_1"><i><font color="Green" title="E56">E8b</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E8:67_2_1_1_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E67">G2n</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E9:67_2_1_1_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E68">G3b</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font>
 ;<br/><a NAME="E10:67_2_1_1_1_1_1_3_1_2_1_1"/>
<font color="Maroon" title="c19">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E11:67_2_1_1_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E69">G3</font></i>: 
<font color="Maroon" title="c19">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:67_2_1_1_1_1_1"><i><font color="Green" title="E56">E8b</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E66">G3dn</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:67_2_1_1_1_1_1_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c19">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E13:67_2_1_1_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E70">G4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i1</font>
 <b>by </b><i><a class="txt" href="#E9:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E68">G3b</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E14:67_2_1_1_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E71">G55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/><a NAME="E15:67_2_1_1_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E72">G20f</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th44</a></i>;<br/><a NAME="E16:67_2_1_1_1_1_1_3_1_2_1_1"/>
<font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th44</a>, <a class="txt" href="#E8:67_2_1_1_1_1_1"><i><font color="Green" title="E56">E8b</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E66">G3dn</font></i></a>, <a class="txt" href="#E5:67_2_1_1_1_1_1"><i><font color="Green" title="E53">E3</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1"><i><font color="Green" title="E54">E4</font></i></a>, <a class="txt" href="#E2:67_2_1_1_1"><i><font color="Green" title="E49">F4</font></i></a></i>;<br/><a NAME="E17:67_2_1_1_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E73">G20</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E14:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E71">G55</font></i></a>, <a class="txt" href="#E5:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E65">G67</font></i></a>, <a class="txt" href="#E15:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E72">G20f</font></i></a>, <a class="ref" href="afinsq_2.html#T42" target="_self" title="AFINSQ_2:th.42">AE222f</a>, <a class="ref" href="afinsq_2.html#T36" target="_self" title="AFINSQ_2:th.36">AE400</a>, <a class="txt" href="#E1:67"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/><a NAME="E18:67_2_1_1_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E74">G21</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E5:67_2_1_1_1_1_1"><i><font color="Green" title="E53">E3</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1"><i><font color="Green" title="E54">E4</font></i></a>, <a class="txt" href="#E8:67_2_1_1_1_1_1"><i><font color="Green" title="E56">E8b</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E66">G3dn</font></i></a></i>;<br/><a NAME="E19:67_2_1_1_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E75">G5</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E17:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E73">G20</font></i></a>, <a class="txt" href="#E18:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E74">G21</font></i></a>, <a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E1:67_2_1_1_1_1_1_3"><i><font color="Green" title="E63">F1</font></i></a></i>;<br/><a NAME="E20:67_2_1_1_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E76">G51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:67_2_1_1_1_1_1"><i><font color="Green" title="E54">E4</font></i></a>, <a class="txt" href="#E11:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E69">G3</font></i></a></i>;<br/><a NAME="E21:67_2_1_1_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E77">G52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:67_2_1_1_1_1_1"><i><font color="Green" title="E54">E4</font></i></a>, <a class="txt" href="#E8:67_2_1_1_1_1_1"><i><font color="Green" title="E56">E8b</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E66">G3dn</font></i></a></i>;<br/><a NAME="E22:67_2_1_1_1_1_1_3_1_2_1_1"/>
<font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:67_2_1_1_1_1_1"><i><font color="Green" title="E53">E3</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1"><i><font color="Green" title="E54">E4</font></i></a>, <a class="txt" href="#E8:67_2_1_1_1_1_1"><i><font color="Green" title="E56">E8b</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E66">G3dn</font></i></a></i>;<br/><b>hence </b><a NAME="E23:67_2_1_1_1_1_1_3_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E67">G2n</font></i></a>, <a class="txt" href="#E13:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E70">G4</font></i></a>, <a class="txt" href="#E19:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E75">G5</font></i></a>, <a class="txt" href="#E20:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E76">G51</font></i></a>, <a class="txt" href="#E21:67_2_1_1_1_1_1_3_1_2_1_1"><i><font color="Green" title="E77">G52</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1_1_1_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:67_2_1_1_1_1_1_3_1_2_1_2"/><i><font color="Green" title="E65">H0</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:67_2_1_1_1_1_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E66">H8</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="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></i>;<br/><a NAME="E3:67_2_1_1_1_1_1_3_1_2_1_2"/><i><font color="Green" title="E67">H12</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><a NAME="E4:67_2_1_1_1_1_1_3_1_2_1_2"/><i><font color="Green" title="E68">G55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/><a NAME="E5:67_2_1_1_1_1_1_3_1_2_1_2"/><i><font color="Green" title="E69">G20f</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th44</a></i>;<br/><a NAME="E6:67_2_1_1_1_1_1_3_1_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T31" target="_self" title="AFINSQ_2:th.31">Th44</a>, <a class="txt" href="#E1:67_2_1_1_1_1_1"><i><font color="Green" title="E51">E1</font></i></a>, <a class="txt" href="#E2:67_2_1_1_1"><i><font color="Green" title="E49">F4</font></i></a></i>;<br/><a NAME="E7:67_2_1_1_1_1_1_3_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T42" target="_self" title="AFINSQ_2:th.42">AE222f</a>, <a class="ref" href="afinsq_2.html#T36" target="_self" title="AFINSQ_2:th.36">AE400</a>, <a class="txt" href="#E1:67"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/><a NAME="E8:67_2_1_1_1_1_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E70">G20</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E69">G20f</font></i></a>, <a class="txt" href="#E3:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E67">H12</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/><a NAME="E9:67_2_1_1_1_1_1_3_1_2_1_2"/><i><font color="Green" title="E71">G2r</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:67_2_1_1_1_1_1"><i><font color="Green" title="E51">E1</font></i></a>, <a class="txt" href="#E5:67_2_1_1_1_1_1"><i><font color="Green" title="E53">E3</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1"><i><font color="Green" title="E54">E4</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><i><font color="Green" title="E72">G2</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></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> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E65">H0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E9:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E71">G2r</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>
;<br/><a NAME="E11:67_2_1_1_1_1_1_3_1_2_1_2"/>
<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>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font>
 <b>by </b><i><a class="txt" href="#E1:67_2_1_1_1"><i><font color="Green" title="E48">D1</font></i></a>, <a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:67_2_1_1_1_1_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E73">G34</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/><a NAME="E13:67_2_1_1_1_1_1_3_1_2_1_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E14:67_2_1_1_1_1_1_3_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E15:67_2_1_1_1_1_1_3_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">g1</font>
 <b>by </b><i><a class="txt" href="#E12:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E73">G34</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E16:67_2_1_1_1_1_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E74">G4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E2:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E66">H8</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E17:67_2_1_1_1_1_1_3_1_2_1_2"/><i><font color="Green" title="E75">G21</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E1:67_2_1_1_1_1_1"><i><font color="Green" title="E51">E1</font></i></a></i>;<br/><a NAME="E18:67_2_1_1_1_1_1_3_1_2_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></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="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E1:67_2_1_1_1_1_1"><i><font color="Green" title="E51">E1</font></i></a></i>;<br/><a NAME="E19:67_2_1_1_1_1_1_3_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E1:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E65">H0</font></i></a>, <a class="txt" href="#E17:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E75">G21</font></i></a>, <a class="txt" href="#E8:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E70">G20</font></i></a>, <a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a>, <a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="txt" href="#E2:67_2_1"><i><font color="Green" title="E47">F3</font></i></a>, <a class="txt" href="#E10:67_2_1_1_1_1_1"><i><font color="Green" title="E58">F4h</font></i></a>, <a class="txt" href="#E4:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E68">G55</font></i></a>, <a class="txt" href="#E5:67"><i><font color="Green" title="E45">A92</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E20:67_2_1_1_1_1_1_3_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E72">G2</font></i></a>, <a class="txt" href="#E16:67_2_1_1_1_1_1_3_1_2_1_2"><i><font color="Green" title="E74">G4</font></i></a>, <a class="txt" href="#E5:67_2_1_1_1_1_1"><i><font color="Green" title="E53">E3</font></i></a>, <a class="ref" href="afinsq_2.html#T6" target="_self" title="AFINSQ_2:th.6">AC500</a>, <a class="txt" href="#E1:67_2_1_1_1_1_1"><i><font color="Green" title="E51">E1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E15:67_2_1_1_1_1_1"/><i><font color="Green" title="E63">R4n</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></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="txt" href="#E5:67_2_1_1_1_1_1"><i><font color="Green" title="E53">E3</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1"><i><font color="Green" title="E54">E4</font></i></a>, <a class="txt" href="#E1:67_2_1_1_1_1_1"><i><font color="Green" title="E51">E1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E16:67_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E64">R4</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/><a NAME="E17:67_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></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="txt" href="#E8:67_2_1_1_1_1_1"><i><font color="Green" title="E56">E8b</font></i></a>, <a class="txt" href="#E1:67_2_1"><i><font color="Green" title="E46">C1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E65">R2</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E15:67_2_1_1_1_1_1"><i><font color="Green" title="E63">R4n</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
;<br/><a NAME="E19:67_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:67_2_1_1_1_1_1"><i><font color="Green" title="E64">R4</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><a NAME="E20:67_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E66">R3</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E21:67_2_1_1_1_1_1"/><i><font color="Green" title="E67">R5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E18:67_2_1_1_1_1_1"><i><font color="Green" title="E65">R2</font></i></a>, <a class="txt" href="#E20:67_2_1_1_1_1_1"><i><font color="Green" title="E66">R3</font></i></a>, <a class="txt" href="#E8:67_2_1_1_1_1_1"><i><font color="Green" title="E56">E8b</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E22:67_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p4">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p4">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="afinsq_2.html#K1" title="AFINSQ_2:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="txt" href="#E21:67_2_1_1_1_1_1"><i><font color="Green" title="E67">R5</font></i></a>, <a class="txt" href="#E5:67_2_1_1_1_1_1"><i><font color="Green" title="E53">E3</font></i></a>, <a class="txt" href="#E6:67_2_1_1_1_1_1"><i><font color="Green" title="E54">E4</font></i></a>, <a class="txt" href="#E16:67_2_1_1_1_1_1"><i><font color="Green" title="E64">R4</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E23:67_2_1_1_1_1_1"/>
 <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#D4" target="_self" title="AFINSQ_2:def.4">AC300</a>, <a class="txt" href="#E9:67_2_1_1_1_1_1"><i><font color="Green" title="E57">E8</font></i></a>, <a class="txt" href="#E13:67_2_1_1_1_1_1"><i><font color="Green" title="E61">E9</font></i></a>, <a class="txt" href="#E14:67_2_1_1_1_1_1"><i><font color="Green" title="E62">E89</font></i></a>, <a class="txt" href="#E3:67_2_1_1_1_1_1"><i><font color="Green" title="E52">E2</font></i></a>, <a class="txt" href="#E5:67_2_1_1_1_1_1"><i><font color="Green" title="E53">E3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_2_1_1_1_1_2"/><i><font color="Green" title="E51">D3</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:67_2_1_1_1_1_2"/><b>then </b>
 <a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:67_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E52">D2</font></i>: 
 <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:67_2_1_1_1_1_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><a NAME="E5:67_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E53">D4</font></i>: 
<font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:67_2_1_1_1_1_2"><i><font color="Green" title="E51">D3</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:67_2_1_1_1_1_2"><i><font color="Green" title="E53">D4</font></i></a></i>
;<br/><b>hence </b><a NAME="E7:67_2_1_1_1_1_2"/>
 <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:67_2_1_1_1_1_2"><i><font color="Green" title="E52">D2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_2"><b>suppose </b></a><a NAME="E1:67_2_1_1_2"/><i><font color="Green" title="E48">D3</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:67_2_1_1_2"/><b>then </b>
 <a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:67_2_1_1_2"/><b>then </b><i><font color="Green" title="E49">D2</font></i>: 
 <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:67_2_1_1_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><a NAME="E5:67_2_1_1_2"/><b>then </b><i><font color="Green" title="E50">D4</font></i>: 
<font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:67_2_1_1_2"><i><font color="Green" title="E48">D3</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:67_2_1_1_2"><i><font color="Green" title="E50">D4</font></i></a></i>
;<br/><b>hence </b><a NAME="E7:67_2_1_1_2"/>
 <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:67_2_1_1_2"><i><font color="Green" title="E49">D2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_2"><b>suppose </b></a><a NAME="E1:67_2_2"/><i><font color="Green" title="E46">C1</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:67_2_2"/><b>then </b><i><font color="Green" title="E47">C2</font></i>: 
 <a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:67_2_2"/>
<font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E4:67_2_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/><a NAME="E5:67_2_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/><a NAME="E6:67_2_2"/><b>then </b><i><font color="Green" title="E48">C7</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:67_2_2"><i><font color="Green" title="E46">C1</font></i></a>, <a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><a NAME="E7:67_2_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <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="#E6:67_2_2"><i><font color="Green" title="E48">C7</font></i></a>, <a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><a NAME="E8:67_2_2"/><b>then </b>
 <a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E9:67_2_2"/><b>then </b><i><font color="Green" title="E49">C8</font></i>: 
 <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E10:67_2_2"/>
<font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E11:67_2_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/><a NAME="E12:67_2_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/><a NAME="E13:67_2_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:67_2_2"><i><font color="Green" title="E46">C1</font></i></a>, <a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><a NAME="E14:67_2_2"/><b>then </b>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span> <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="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">AC200</a></i>;<br/><b>hence </b><a NAME="E15:67_2_2"/>
 <a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">B1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K8" title="AFINSQ_2:func.8">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K11" title="AFINSQ_2:func.11">SubXFinS</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">B2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:67_2_2"><i><font color="Green" title="E47">C2</font></i></a>, <a class="txt" href="#E9:67_2_2"><i><font color="Green" title="E49">C8</font></i></a>, <a class="ref" href="afinsq_1.html#T18" title="AFINSQ_1:th.18">AFINSQ_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
