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

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:92_1_2"/><i><font color="Green" title="E50">A25</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font> ) )
 <b>and </b><br/><a NAME="E2:92_1_2"/><i><font color="Green" title="E51">A26</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>] <b>means </b>for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  ex <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b2">k</font> &amp; $1 is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> $1 &amp; <font color="Olive" title="b4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k2</font> ) holds <br/>for <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b3">l</font>, <font color="Olive" title="b4">m</font>, <font color="Olive" title="b5">k1</font>, <font color="Olive" title="b6">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">l</font> &amp; <font color="Olive" title="b3">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">m</font> &amp; <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b5">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">l</font> &amp; <font color="Olive" title="b6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b5">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">k2</font> ) holds <br/><font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<a NAME="E3:92_1_2"/><i><font color="Green" title="E52">A27</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<a NAME="E4:92_1_2"/><i><font color="Green" title="E53">A28</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:92_1_2_1"/><i><font color="Green" title="E54">A29</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">p</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c7">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>given </b><font color="Maroon" title="c8">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E3:92_1_2_1"/><i><font color="Green" title="E55">A30</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">k</font>
 ;<br/>

<b>assume </b><a NAME="E4:92_1_2_1"/><i><font color="Green" title="E56">A31</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font> ) )
 ;<br/>

<b>let </b><font color="Maroon" title="c9">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E5:92_1_2_1"/><i><font color="Green" title="E57">A32</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font> ) )
 ;<br/>

<a NAME="E6:92_1_2_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
;<br/>
<a NAME="E7:92_1_2_1"/><b>then </b><i><font color="Green" title="E58">A33</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" target="_self" title="FINSEQ_1:def.8">Def8</a></i>;<br/>
<a NAME="E8:92_1_2_1"/><i><font color="Green" title="E59">A34</font></i>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; ( for <font color="Olive" title="b2">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Olive" title="b2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> holds <br/><font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:92_1_2_1_1"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="ref" href="finseq_1.html#T50" target="_self" title="FINSEQ_1:th.50">Th50</a></i>;<br/>
<a NAME="E2:92_1_2_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" target="_self" title="FINSEQ_1:def.4">Def4</a></i>;<br/>
<a NAME="E3:92_1_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" target="_self" title="FINSEQ_1:th.55">Th55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">m</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">m</font>
;<br/>

<b>thus </b><a NAME="E5:92_1_2_1_1"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/>

<b>thus </b><a NAME="E6:92_1_2_1_1"/>
for <font color="Olive" title="b1">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> holds <br/><font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">l</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_1_1"/><i><font color="Green" title="E60">A35</font></i>: 
( <font color="Maroon" title="c11">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c11">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:92_1_2_1_1_1"/><i><font color="Green" title="E61">A36</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:92_1_2_1_1_1"/><i><font color="Green" title="E62">A37</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">k</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_1_1"><i><font color="Green" title="E61">A36</font></i></a></i>;<br/>
<a NAME="E6:92_1_2_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1_1_1"><i><font color="Green" title="E62">A37</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>
<a NAME="E7:92_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>;<br/>
<a NAME="E8:92_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E63">A38</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>;<br/>
<a NAME="E9:92_1_2_1_1_1"/>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1_2_1_1_1_1"/>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="finseq_1.html#E7:92_1_2_1"><i><font color="Green" title="E58">A33</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D8" target="_self" title="FINSEQ_1:def.8">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E3:92_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_1_1"><i><font color="Green" title="E60">A35</font></i></a>, <a class="txt" href="finseq_1.html#E3:92_1_2_1_1_1"><i><font color="Green" title="E61">A36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:92_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E8:92_1_2_1_1_1"><i><font color="Green" title="E63">A38</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E11:92_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>;<br/>
<a NAME="E12:92_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E64">A39</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>;<br/>
<font color="Maroon" title="c10">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T59" target="_self" title="FINSEQ_1:th.59">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
;<br/>
<b>hence </b><a NAME="E14:92_1_2_1_1_1"/>
<font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="finseq_1.html#E3:92_1_2_1_1_1"><i><font color="Green" title="E61">A36</font></i></a>, <a class="txt" href="finseq_1.html#E12:92_1_2_1_1_1"><i><font color="Green" title="E64">A39</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">m</font> = <font color="Maroon" title="c6">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E10:92_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</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="E2:92_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">X</font> <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="finseq_1.html#E5:92_1_2_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="txt" href="finseq_1.html#E32"><i><font color="Green" title="E20">Lm4</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<a NAME="E3:92_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></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="finseq_1.html#E4:92_1_2_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>then </b> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E32"><i><font color="Green" title="E20">Lm4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
<br/>.= 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
;<br/>
<b>hence </b><a NAME="E5:92_1_2_1_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c11">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:92_1_2_1"/><i><font color="Green" title="E60">A40</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E13:92_1_2_1"/>
 ex <font color="Olive" title="b1">q'</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; ( for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Olive" title="b1">q'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">m</font>) ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S2" target="_self" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<b>then consider </b><font color="Maroon" title="c12">q'</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E15:92_1_2_1"/><i><font color="Green" title="E61">A41</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Maroon" title="c12">q'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> ) )
 ;<br/>
<a NAME="E16:92_1_2_1"/>
<font color="Maroon" title="c12">q'</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:92_1_2_1_3_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">q'</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:92_1_2_1_3_1"/><i><font color="Green" title="E62">A42</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">q'</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:92_1_2_1_3_1"/><i><font color="Green" title="E63">A43</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q'</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_3_1"><i><font color="Green" title="E62">A42</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_3_1"><i><font color="Green" title="E62">A42</font></i></a></i>;<br/><a NAME="E6:92_1_2_1_3_1"/><i><font color="Green" title="E64">A44</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_3_1"><i><font color="Green" title="E63">A43</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/><a NAME="E7:92_1_2_1_3_1"/>
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E8:92_1_2_1_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:92_1_2_1_3_1"><i><font color="Green" title="E64">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:92_1_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><a NAME="E10:92_1_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_2_1"><i><font color="Green" title="E60">A40</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/><a NAME="E11:92_1_2_1_3_1"/><b>then </b><i><font color="Green" title="E65">A45</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:92_1_2_1_3_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" target="_self" title="FINSEQ_1:def.4">Def4</a></i>;<br/><a NAME="E13:92_1_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="finseq_1.html#E11:92_1_2_1_3_1"><i><font color="Green" title="E65">A45</font></i></a></i>;<br/><b>hence </b><a NAME="E14:92_1_2_1_3_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="txt" href="finseq_1.html#E3:92_1_2_1_3_1"><i><font color="Green" title="E62">A42</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_3_1"><i><font color="Green" title="E63">A43</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:92_1_2_1_3"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">q'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:92_1_2_1_3"/>
<font color="Maroon" title="c12">q'</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" target="_self" title="FINSEQ_1:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c12">q'</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E18:92_1_2_1"/><i><font color="Green" title="E62">A46</font></i>: 
<font color="Maroon" title="c12">q'</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_4"><b>proof </b></a><div class="add">

<i><font color="Green" title="E63">A47</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_2_1"><i><font color="Green" title="E60">A40</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q'</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
;<br/>
<a NAME="E2:92_1_2_1_4"/><i><font color="Green" title="E64">A48</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">q'</font> holds <br/><font color="Maroon" title="c12">q'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_4_2"/>
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">q'</font>
 ;<br/>

<a NAME="E2:92_1_2_1_4_2"/><b>then </b>
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:92_1_2_1_4_2"/>
<font color="Maroon" title="c12">q'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:92_1_2_1_4"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> holds <br/><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q'</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_4_3"/>
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 ;<br/>

<a NAME="E2:92_1_2_1_4_3"/><b>then </b><i><font color="Green" title="E65">A49</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" target="_self" title="FINSEQ_1:def.8">Def8</a>, <a class="ref" href="finseq_1.html#T4" target="_self" title="FINSEQ_1:th.4">Th4</a></i>;<br/>
<a NAME="E3:92_1_2_1_4_3"/><b>then </b><i><font color="Green" title="E66">A50</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:92_1_2_1_4_3"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q'</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_4_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1_2_1_4_3_1"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q'</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font>
 ;<br/>

<a NAME="E2:92_1_2_1_4_3_1"/><b>then </b><i><font color="Green" title="E67">A51</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_2_1"><i><font color="Green" title="E60">A40</font></i></a>, <a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2_1_4_3"><i><font color="Green" title="E65">A49</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:92_1_2_1_4_3_1"/><i><font color="Green" title="E68">A52</font></i>: 
( <font color="Maroon" title="c15">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; ( for <font color="Olive" title="b1">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> holds <br/><font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">d</font> ) )
 <b>by </b><i><a class="txt" href="finseq_1.html#E8:92_1_2_1"><i><font color="Green" title="E59">A34</font></i></a></i>;<br/>
<a NAME="E5:92_1_2_1_4_3_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_4_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:92_1_2_1_4_3_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:92_1_2_1_4_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" target="_self" title="FINSEQ_1:th.55">Th55</a></i>;<br/>
<a NAME="E3:92_1_2_1_4_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1_2_1_4_3_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="finseq_1.html#E5:92_1_2_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="ref" href="finseq_1.html#T44" target="_self" title="FINSEQ_1:th.44">Th44</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c16">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:92_1_2_1_4_3_1"/><i><font color="Green" title="E69">A53</font></i>: 
( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:92_1_2_1_4_3_1"/><i><font color="Green" title="E70">A54</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E7:92_1_2_1_4_3_1"><i><font color="Green" title="E69">A53</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E7:92_1_2_1_4_3_1"><i><font color="Green" title="E69">A53</font></i></a></i>;<br/>
<a NAME="E10:92_1_2_1_4_3_1"/><i><font color="Green" title="E71">A55</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E8:92_1_2_1_4_3_1"><i><font color="Green" title="E70">A54</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>
<a NAME="E11:92_1_2_1_4_3_1"/><b>then </b><i><font color="Green" title="E72">A56</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_4_3_1"><i><font color="Green" title="E67">A51</font></i></a>, <a class="txt" href="finseq_1.html#E7:92_1_2_1_4_3_1"><i><font color="Green" title="E69">A53</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E12:92_1_2_1_4_3_1"/><i><font color="Green" title="E73">A57</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> &amp; <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_4_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:92_1_2_1_4_3_1_2"/><i><font color="Green" title="E74">A58</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" target="_self" title="FINSEQ_1:def.4">Def4</a></i>;<br/>
<a NAME="E2:92_1_2_1_4_3_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E10:92_1_2_1_4_3_1"><i><font color="Green" title="E71">A55</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:92_1_2_1_4_3_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span>
 
;<br/>
<a NAME="E4:92_1_2_1_4_3_1_2"/><b>then </b><i><font color="Green" title="E75">A59</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<a NAME="E5:92_1_2_1_4_3_1_2"/><b>then </b>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">q</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:92_1_2_1_4_3_1_2"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_4_3_1_2"><i><font color="Green" title="E74">A58</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>

<b>thus </b><a NAME="E7:92_1_2_1_4_3_1_2"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E5:92_1_2_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_4_3_1_2"><i><font color="Green" title="E75">A59</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E14:92_1_2_1_4_3_1"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">d</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_4_3_1"><i><font color="Green" title="E67">A51</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_4_3_1"><i><font color="Green" title="E68">A52</font></i></a>, <a class="txt" href="finseq_1.html#E12:92_1_2_1_4_3_1"><i><font color="Green" title="E73">A57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:92_1_2_1_4_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_1.html#E5:92_1_2_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_4_3_1"><i><font color="Green" title="E68">A52</font></i></a>, <a class="txt" href="finseq_1.html#E7:92_1_2_1_4_3_1"><i><font color="Green" title="E69">A53</font></i></a>, <a class="txt" href="finseq_1.html#E11:92_1_2_1_4_3_1"><i><font color="Green" title="E72">A56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:92_1_2_1_4_3"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q'</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_4_3"><i><font color="Green" title="E66">A50</font></i></a>, <a class="ref" href="finseq_1.html#T57" target="_self" title="FINSEQ_1:th.57">Th57</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:92_1_2_1_4"/>
<font color="Maroon" title="c12">q'</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_4"><i><font color="Green" title="E63">A47</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2_1_4"><i><font color="Green" title="E64">A48</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:92_1_2_1"/>
<font color="Maroon" title="c12">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:92_1_2_1_5"/><i><font color="Green" title="E63">A60</font></i>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c8">k</font>
;<br/>

<b>thus </b><a NAME="E1:92_1_2_1_5_1"/>
<font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1"><i><font color="Green" title="E55">A30</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:92_1_2_1_5"/><i><font color="Green" title="E64">A61</font></i>: 
( <font color="Maroon" title="c5">p</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:92_1_2_1_5_2"/>
<font color="Maroon" title="c5">p</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="ref" href="finseq_1.html#T50" target="_self" title="FINSEQ_1:th.50">Th50</a></i>;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:92_1_2_1_5_2_1"/><i><font color="Green" title="E65">A62</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1_2_1_5_2_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:92_1_2_1_5_2_1_1"/><i><font color="Green" title="E66">A63</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_2_1_1"/><i><font color="Green" title="E67">A64</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_2_1_1"><i><font color="Green" title="E66">A63</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_2_1_1"><i><font color="Green" title="E66">A63</font></i></a></i>;<br/>
<a NAME="E6:92_1_2_1_5_2_1_1"/><i><font color="Green" title="E68">A65</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1_5_2_1_1"><i><font color="Green" title="E67">A64</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>
<a NAME="E7:92_1_2_1_5_2_1_1"/><i><font color="Green" title="E69">A66</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:92_1_2_1_5_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">y</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1_5_2_1_1"><i><font color="Green" title="E67">A64</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_2_1_1_1"/>
<font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:92_1_2_1_5_2_1_1"><i><font color="Green" title="E68">A65</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>

<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
;<br/>
<b>hence </b><a NAME="E4:92_1_2_1_5_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:92_1_2_1_5_2_1_1"/><i><font color="Green" title="E70">A67</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_2_1_1"><i><font color="Green" title="E66">A63</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>;<br/>
<a NAME="E9:92_1_2_1_5_2_1_1"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59" target="_self" title="FINSEQ_1:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E10:92_1_2_1_5_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="finseq_1.html#E7:92_1_2_1_5_2_1_1"><i><font color="Green" title="E69">A66</font></i></a>, <a class="txt" href="finseq_1.html#E8:92_1_2_1_5_2_1_1"><i><font color="Green" title="E70">A67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E66">A68</font></i>: <font color="Maroon" title="c7">X</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="ref" href="finseq_1.html#T44" target="_self" title="FINSEQ_1:th.44">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
;<br/>
<a NAME="E3:92_1_2_1_5_2_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> iff <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:92_1_2_1_5_2_1_3"/>
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_2_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1_2_1_5_2_1_3_1"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:92_1_2_1_5_2_1_3_1"/><b>then </b>
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:92_1_2_1_5_2_1_3_1"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:92_1_2_1_5_2_1_3"/><i><font color="Green" title="E67">A69</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font>
 ;<br/>

<a NAME="E3:92_1_2_1_5_2_1_3"/><b>then </b><i><font color="Green" title="E68">A70</font></i>: 
not <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_1"><i><font color="Green" title="E65">A62</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_2_1_3"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_5_2_1_3"><i><font color="Green" title="E67">A69</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:92_1_2_1_5_2_1_3"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_2_1_3"><i><font color="Green" title="E68">A70</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:92_1_2_1_5_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_5_2_1"><i><font color="Green" title="E66">A68</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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

<b>thus </b><a NAME="E3:92_1_2_1_5_2"/>
for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">l</font>, <font color="Maroon" title="c15">m</font>, <font color="Maroon" title="c16">k1</font>, <font color="Maroon" title="c17">k2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_5_2_2"/><i><font color="Green" title="E65">A71</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">l</font> &amp; <font color="Maroon" title="c14">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">l</font> &amp; <font color="Maroon" title="c17">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font> )
 ;<br/>

<a NAME="E2:92_1_2_1_5_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">l</font> &amp; <font color="Maroon" title="c14">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon" title="c14">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon" title="c14">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<a NAME="E5:92_1_2_1_5_2_2"/><b>then </b><i><font color="Green" title="E66">A72</font></i>: 
<font color="Maroon" title="c16">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">l</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_2"><i><font color="Green" title="E65">A71</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>;<br/>
<a NAME="E6:92_1_2_1_5_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_2"><i><font color="Green" title="E65">A71</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>
<a NAME="E8:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<a NAME="E9:92_1_2_1_5_2_2"/><b>then </b><i><font color="Green" title="E67">A73</font></i>: 
<font color="Maroon" title="c17">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_2"><i><font color="Green" title="E65">A71</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>;<br/>
<a NAME="E10:92_1_2_1_5_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E11:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_2"><i><font color="Green" title="E65">A71</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>;<br/>
<a NAME="E13:92_1_2_1_5_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">l</font> &amp; <font color="Maroon" title="c14">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_2"><i><font color="Green" title="E65">A71</font></i></a>, <a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E14:92_1_2_1_5_2_2"/>
<font color="Maroon" title="c16">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">k2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="finseq_1.html#E5:92_1_2_1_5_2_2"><i><font color="Green" title="E66">A72</font></i></a>, <a class="txt" href="finseq_1.html#E9:92_1_2_1_5_2_2"><i><font color="Green" title="E67">A73</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:92_1_2_1_5"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; ( for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:92_1_2_1_5_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:92_1_2_1_5_3_1"/><i><font color="Green" title="E65">A74</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1_2_1_5_3_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:92_1_2_1_5_3_1_1"/><i><font color="Green" title="E66">A75</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_3_1_1"/><i><font color="Green" title="E67">A76</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_3_1_1"><i><font color="Green" title="E66">A75</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self" title="FINSEQ_1:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_3_1_1"><i><font color="Green" title="E66">A75</font></i></a></i>;<br/>
<a NAME="E6:92_1_2_1_5_3_1_1"/><i><font color="Green" title="E68">A77</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1_5_3_1_1"><i><font color="Green" title="E67">A76</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>
<a NAME="E7:92_1_2_1_5_3_1_1"/><i><font color="Green" title="E69">A78</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font></span>)</span> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_3_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:92_1_2_1_5_3_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">y</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1_5_3_1_1"><i><font color="Green" title="E67">A76</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_3_1_1_1"/>
<font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:92_1_2_1_5_3_1_1"><i><font color="Green" title="E68">A77</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>

<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" target="_self" title="FINSEQ_1:th.35">Th35</a></i>
;<br/>
<b>hence </b><a NAME="E4:92_1_2_1_5_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font></span>)</span> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:92_1_2_1_5_3_1_1"/><i><font color="Green" title="E70">A79</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E18:92_1_2_1"><i><font color="Green" title="E62">A46</font></i></a>, <a class="txt" href="finseq_1.html#E3:92_1_2_1_5_3_1_1"><i><font color="Green" title="E66">A75</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self" title="FINSEQ_1:def.7">Def7</a></i>;<br/>
<a NAME="E9:92_1_2_1_5_3_1_1"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E18:92_1_2_1"><i><font color="Green" title="E62">A46</font></i></a>, <a class="ref" href="finseq_1.html#T59" target="_self" title="FINSEQ_1:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E10:92_1_2_1_5_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_1.html#E5:92_1_2_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="txt" href="finseq_1.html#E18:92_1_2_1"><i><font color="Green" title="E62">A46</font></i></a>, <a class="txt" href="finseq_1.html#E7:92_1_2_1_5_3_1_1"><i><font color="Green" title="E69">A78</font></i></a>, <a class="txt" href="finseq_1.html#E8:92_1_2_1_5_3_1_1"><i><font color="Green" title="E70">A79</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E66">A80</font></i>: <font color="Maroon" title="c7">X</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E5:92_1_2_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="txt" href="finseq_1.html#E18:92_1_2_1"><i><font color="Green" title="E62">A46</font></i></a>, <a class="ref" href="finseq_1.html#T44" target="_self" title="FINSEQ_1:th.44">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self" title="FINSEQ_1:th.56">Th56</a></i>
;<br/>
<a NAME="E3:92_1_2_1_5_3_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> iff <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:92_1_2_1_5_3_1_3"/>
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_3_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1_2_1_5_3_1_3_1"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:92_1_2_1_5_3_1_3_1"/><b>then </b>
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:92_1_2_1_5_3_1_3_1"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:92_1_2_1_5_3_1_3"/><i><font color="Green" title="E67">A81</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font>
 ;<br/>

<a NAME="E3:92_1_2_1_5_3_1_3"/><b>then </b><i><font color="Green" title="E68">A82</font></i>: 
not <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_1"><i><font color="Green" title="E65">A74</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_3_1_3"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_5_3_1_3"><i><font color="Green" title="E67">A81</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:92_1_2_1_5_3_1_3"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_3_1_3"><i><font color="Green" title="E68">A82</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:92_1_2_1_5_3_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_5_3_1"><i><font color="Green" title="E66">A80</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:92_1_2_1_5_3"/>
for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Olive" title="b3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">l</font>, <font color="Maroon" title="c15">m</font>, <font color="Maroon" title="c16">k1</font>, <font color="Maroon" title="c17">k2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_5_3_2"/><i><font color="Green" title="E65">A83</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">l</font> &amp; <font color="Maroon" title="c14">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">l</font> &amp; <font color="Maroon" title="c17">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font> )
 ;<br/>

<a NAME="E2:92_1_2_1_5_3_2"/><b>then </b><i><font color="Green" title="E66">A84</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">l</font> &amp; <font color="Maroon" title="c14">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E12:92_1_2_1"><i><font color="Green" title="E60">A40</font></i></a>, <a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:92_1_2_1_5_3_2"/>
( <font color="Maroon" title="c14">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2_1_5_3_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:92_1_2_1_5_3_2_1"/>
<font color="Maroon" title="c14">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green" title="E65">A83</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E2:92_1_2_1_5_3_2_1"/>
<font color="Maroon" title="c14">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green" title="E65">A83</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>

<a NAME="E3:92_1_2_1_5_3_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green" title="E65">A83</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1_2_1_5_3_2_1"/>
<font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green" title="E65">A83</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self" title="FINSEQ_1:th.3">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:92_1_2_1_5_3_2"/><b>then </b>
( <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">l</font> &amp; <font color="Maroon" title="c17">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E15:92_1_2_1"><i><font color="Green" title="E61">A41</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green" title="E65">A83</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:92_1_2_1_5_3_2"/>
<font color="Maroon" title="c16">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">k2</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E5:92_1_2_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2_1_5_3_2"><i><font color="Green" title="E66">A84</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:92_1_2_1_5"/>
<font color="Maroon" title="c12">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1"><i><font color="Green" title="E54">A29</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2_1_5"><i><font color="Green" title="E63">A60</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2_1_5"><i><font color="Green" title="E64">A61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:92_1_2_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E18:92_1_2_1"><i><font color="Green" title="E62">A46</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:92_1_2"/>
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_1.html#S3" target="_self" title="FINSEQ_1:sch.3">FINSEQ_1:sch 3</a>(<a class="txt" href="finseq_1.html#E3:92_1_2"><i><font color="Green" title="E52">A27</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2"><i><font color="Green" title="E53">A28</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:92_1_2"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:92"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2"><i><font color="Green" title="E51">A26</font></i></a></i>;<br/>


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