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

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> be    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:92_1_2"/><i><font color="Green">E87</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) )
 <b>and </b><br/><a NAME="E2:92_1_2"/><i><font color="Green">E88</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="finseq_1.html#NM1">FinSequence</a>] <b>means </b>for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st  ex <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Olive">k</font> &amp; <font color="Maroon">a<sub>1</sub></font> is    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) holds <br/>for <font color="Olive">q</font> being   <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="relat_1.html#K2">rng</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">q</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) holds <br/><font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:92_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<a NAME="E4:92_1_2"/><i><font color="Green">E125</font></i>: 
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Olive">x</font></span><a href="finseq_1.html#K9" target="_self">*&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">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:92_1_2_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">p</font>]
 ;<br/>

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>given </b><font color="Maroon">k</font> being  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><a NAME="E2:92_1_2_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font>
 ;<br/>

<b>assume </b><a NAME="E3:92_1_2_1"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> is    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:92_1_2_1"/><i><font color="Green">E129</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k2</font> ) )
 ;<br/>

<a NAME="E5:92_1_2_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> 1
 
;<br/>
<a NAME="E6:92_1_2_1"/><b>then </b><i><font color="Green">E130</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:92_1_2_1"/><i><font color="Green">E131</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">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" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> is    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E2:92_1_2_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:92_1_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" target="_self">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

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

<b>thus </b><a NAME="E6:92_1_2_1_1"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">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">l</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_1_1"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:92_1_2_1_1_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:92_1_2_1_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a></i>;<br/>
<a NAME="E6:92_1_2_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self">Def3</a>, </i>;<br/>
<a NAME="E7:92_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:92_1_2_1_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E9:92_1_2_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</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">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><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#E15"><i><font color="Green">Lemma60</font></i></a>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</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">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">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">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E12:92_1_2_1_1_1"/><b>then </b><i><font color="Green">E136</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> )
 
<b>by </b><i/>;<br/>
<font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E14:92_1_2_1_1_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="finseq_1.html#T14" target="_self">Th14</a></i>;<br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E9:92_1_2_1"/>
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<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" target="_self">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:92_1_2_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E3:92_1_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<b>then </b>0 = 
 <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</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">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:92_1_2_1"/><i><font color="Green">E137</font></i>: 
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E13:92_1_2_1"/>
 ex <font color="Olive">q'</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Olive">q'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>) ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S2" target="_self">FINSEQ_1:sch 2</a>();<br/></i>
<b>then consider </b><font color="Maroon">q'</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E15:92_1_2_1"/><i><font color="Green">E138</font></i>: 
(  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">q'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> ) )
 ;<br/>
<a NAME="E16:92_1_2_1"/>
<font color="Maroon">q'</font> is    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:92_1_2_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q'</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:92_1_2_1_3_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:92_1_2_1_3_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q'</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self">Def3</a>, </i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:92_1_2_1_3_1"/><i><font color="Green">E141</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, , </i>;<br/><a NAME="E7:92_1_2_1_3_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E8:92_1_2_1_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:92_1_2_1_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E10:92_1_2_1_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/><a NAME="E11:92_1_2_1_3_1"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:92_1_2_1_3_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><a NAME="E13:92_1_2_1_3_1"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E14:92_1_2_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</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">rng</a> <font color="Maroon">q'</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:92_1_2_1_3"/>
<font color="Maroon">q'</font> is    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">q'</font> as    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E18:92_1_2_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">q'</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E144</font></i>:  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font> = 
 <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma60</font></i></a>, </i>
<br/>.= 
 <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q'</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p3"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>
;<br/>
<a NAME="E2:92_1_2_1_4"/><i><font color="Green">E145</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q'</font> holds <br/><font color="Maroon">q'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_4_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q'</font>
 ;<br/>

<a NAME="E2:92_1_2_1_4_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:92_1_2_1_4_2"/>
<font color="Maroon">q'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:92_1_2_1_4"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> holds <br/><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q'</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">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">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_4_3"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 ;<br/>

<a NAME="E2:92_1_2_1_4_3"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:92_1_2_1_4_3"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:92_1_2_1_4_3"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q'</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">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">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q'</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:92_1_2_1_4_3_1"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma60</font></i></a>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:92_1_2_1_4_3_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:92_1_2_1_4_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">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">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:92_1_2_1_4_3_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E3:92_1_2_1_4_3_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1_2_1_4_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="txt" href="finseq_1.html#E1:92"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:92_1_2_1_4_3_1"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:92_1_2_1_4_3_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E10:92_1_2_1_4_3_1"/><i><font color="Green">E152</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T21" target="_self">Th21</a>, </i>;<br/>
<a NAME="E11:92_1_2_1_4_3_1"/><b>then </b><i><font color="Green">E153</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E12:92_1_2_1_4_3_1"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E155</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:92_1_2_1_4_3_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">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" target="_self">len</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 
;<br/>
<a NAME="E4:92_1_2_1_4_3_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:92_1_2_1_4_3_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:92_1_2_1_4_3_1_2"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T23" target="_self">Th23</a></i>;<br/>

<b>thus </b><a NAME="E7:92_1_2_1_4_3_1_2"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E14:92_1_2_1_4_3_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T17" target="_self">Th17</a>, <a class="txt" href="finseq_1.html#E28"><i><font color="Green">Lemma73</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="ref" href="finseq_1.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:92_1_2_1_4_3"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q'</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:92_1_2_1_4"/>
<font color="Maroon">q'</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T14" target="_self">Th14</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:92_1_2_1"/>
<font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E157</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Olive">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>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_1"/>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:92_1_2_1_5"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">p</font> is    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">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">p</font> is    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T5" target="_self">Th5</a></i>;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</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">E161</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">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">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:92_1_2_1_5_2_1_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_2_1_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:92_1_2_1_5_2_1_1"/><i><font color="Green">E164</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:92_1_2_1_5_2_1_1"/><i><font color="Green">E165</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&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">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_2_1_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T17" target="_self">Th17</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>

<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><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" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&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">E166</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:92_1_2_1_5_2_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E10:92_1_2_1_5_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T21" target="_self">Th21</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E167</font></i>: <font color="Maroon">X</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="finseq_1.html#E1:92"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>
;<br/>
<a NAME="E3:92_1_2_1_5_2_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> iff <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:92_1_2_1_5_2_1_3"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:92_1_2_1_5_2_1_3_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">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">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">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">E168</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:92_1_2_1_5_2_1_3"/><b>then </b><i><font color="Green">E169</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_2_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:92_1_2_1_5_2_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_0.html#D4">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">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">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">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">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">l</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">k1</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">k2</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_5_2_2"/><i><font color="Green">E170</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:92_1_2_1_5_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:92_1_2_1_5_2_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:92_1_2_1_5_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:92_1_2_1_5_2_2"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:92_1_2_1_5_2_2"/>
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E11:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E13:92_1_2_1_5_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:92_1_2_1_5_2_2"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font>
 <b>by </b><i>, , </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">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; ( for <font color="Olive">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">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">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</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">E173</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">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">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:92_1_2_1_5_3_1_1"/><i><font color="Green">E174</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_3_1_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:92_1_2_1_5_3_1_1"/><i><font color="Green">E176</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:92_1_2_1_5_3_1_1"/><i><font color="Green">E177</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&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">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_3_1_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T17" target="_self">Th17</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>

<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><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" target="_self">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&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">E178</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self">Def3</a>, , </i>;<br/>
<a NAME="E9:92_1_2_1_5_3_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E10:92_1_2_1_5_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T21" target="_self">Th21</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E180</font></i>: <font color="Maroon">X</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a>, <a class="txt" href="finseq_1.html#E1:92"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>
;<br/>
<a NAME="E3:92_1_2_1_5_3_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> iff <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:92_1_2_1_5_3_1_3"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:92_1_2_1_5_3_1_3_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">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">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">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">E181</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:92_1_2_1_5_3_1_3"/><b>then </b><i><font color="Green">E183</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_3_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:92_1_2_1_5_3_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_0.html#D4">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">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">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">l</font>, <font color="Olive">m</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">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">l</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">k1</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">k2</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_5_3_2"/><i><font color="Green">E184</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:92_1_2_1_5_3_2"/><b>then </b><i><font color="Green">E185</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma60</font></i></a>, <a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:92_1_2_1_5_3_2"/>
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">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">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E2:92_1_2_1_5_3_2_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>

<a NAME="E3:92_1_2_1_5_3_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1_2_1_5_3_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:92_1_2_1_5_3_2"/><b>then </b>
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E16"><i><font color="Green">Lemma61</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:92_1_2_1_5_3_2"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font>
 <b>by </b><i>, </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">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:92_1_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:92_1_2"/>
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_1.html#S3" target="_self">FINSEQ_1:sch 3</a>(, );<br/></i>
<b>hence </b><a NAME="E6:92_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , </i>;<br/>


</div>
