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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">E50</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> ) )
 <b>and </b><br/><a NAME="E2:92_1_2"/><i><font color="Green">E51</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="finseq_1.html#NM1">FinSequence</a>] means for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st ex <font color="Olive">b<sub>2</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></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">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></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">b<sub>4</sub></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">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></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">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>5</sub></font> ) holds <br/>for <font color="Olive">b<sub>2</sub></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">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></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">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>6</sub></font> ) holds <br/><font color="Olive">b<sub>2</sub></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">E52</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">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></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">b<sub>2</sub></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">c<sub>5</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:92_1_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>given </b><font color="Maroon">c<sub>8</sub></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">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:92_1_2_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>5</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></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">c<sub>5</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> ) )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></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">E57</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></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">E58</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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E7:92_1_2_1"/><i><font color="Green">E59</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></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="txt" href="finseq_1.html#E3:92_1_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T50" target="_self">Th50</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">c<sub>6</sub></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><a class="ref" href="finseq_1.html#D4" target="_self">Def4</a></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">c<sub>6</sub></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#T55" target="_self">Th55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">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">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E5:92_1_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>thus </b><a NAME="E6:92_1_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></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">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></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">E61</font></i>: 
( <font color="Maroon">c<sub>12</sub></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">c<sub>5</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1"><i><font color="Green">E56</font></i></a>, <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">E62</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>5</sub></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">c<sub>6</sub></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#E3:92_1_2_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_1_1"><i><font color="Green">E61</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">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></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">c<sub>6</sub></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#E4:92_1_2_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</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">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></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">c<sub>6</sub></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#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E8:92_1_2_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E9:92_1_2_1_1_1"/>
<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></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">c<sub>13</sub></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">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>13</sub></font> = 
<span class="p1"><a href="finseq_1.html#K9" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></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><a class="txt" href="finseq_1.html#E6:92_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D8" target="_self">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E3:92_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="finseq_1.html#E3:92_1_2_1_1_1"><i><font color="Green">E61</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">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E8:92_1_2_1_1_1"><i><font color="Green">E63</font></i></a>, <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">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></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">c<sub>6</sub></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#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E12:92_1_2_1_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>5</sub></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">c<sub>6</sub></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#T35" target="_self">Th35</a></i>;<br/>
<font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T59" target="_self">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>5</sub></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">c<sub>6</sub></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#T56" target="_self">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>5</sub></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">c<sub>6</sub></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#T35" target="_self">Th35</a></i>
;<br/>
<b>hence </b><a NAME="E14:92_1_2_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="finseq_1.html#E3:92_1_2_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_1.html#E12:92_1_2_1_1_1"><i><font color="Green">E64</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E9:92_1_2_1"/>
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:92_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="finseq_1.html#E32"><i><font color="Green">Lemma19</font></i></a>, <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">c<sub>5</sub></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">c<sub>6</sub></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="txt" href="finseq_1.html#E3:92_1_2_1"><i><font color="Green">E56</font></i></a>, <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">c<sub>5</sub></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">c<sub>6</sub></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#E32"><i><font color="Green">Lemma19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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#T35" target="_self">Th35</a></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">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self">Th56</a></i>
;<br/>
<b>hence </b><a NAME="E5:92_1_2_1_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>11</sub></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">E60</font></i>: 
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E12:92_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>11</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></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">c<sub>12</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E14:92_1_2_1"/><i><font color="Green">E61</font></i>: 
(  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>11</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<a NAME="E15:92_1_2_1"/>
<font color="Maroon">c<sub>12</sub></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">c<sub>13</sub></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">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></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">E62</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></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">E63</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_3_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_3_1"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E6:92_1_2_1_3_1"/><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E14:92_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_3_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E7:92_1_2_1_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></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">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:92_1_2_1_3_1"><i><font color="Green">E64</font></i></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">c<sub>15</sub></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">c<sub>11</sub></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">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E11:92_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E11:92_1_2_1_3_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <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#D4" target="_self">Def4</a></i>;<br/><a NAME="E13:92_1_2_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></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#E11:92_1_2_1_3_1"><i><font color="Green">E65</font></i></a></i>;<br/><b>hence </b><a NAME="E14:92_1_2_1_3_1"/>
<font color="Maroon">c<sub>13</sub></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#E14:92_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_1.html#E3:92_1_2_1_3_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_3_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:92_1_2_1_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></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">c<sub>12</sub></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#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="finseq_1.html#M2" target="_self">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E17:92_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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">E63</font></i>:  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>9</sub></font> = 
 <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="finseq_1.html#E11:92_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</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">c<sub>12</sub></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">c<sub>6</sub></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#E14:92_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#T56" target="_self">Th56</a></i>
;<br/>
<a NAME="E2:92_1_2_1_4"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>12</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_4_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E2:92_1_2_1_4_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E14:92_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:92_1_2_1_4_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E14:92_1_2_1"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:92_1_2_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> holds <br/><font color="Maroon">c<sub>9</sub></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">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:92_1_2_1_4_3"/>
<font color="Maroon">c<sub>14</sub></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">c<sub>6</sub></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">E65</font></i>: 
<font color="Maroon">c<sub>14</sub></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><a class="ref" href="finseq_1.html#D8" target="_self">Def8</a>, <a class="ref" href="finseq_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:92_1_2_1_4_3"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>9</sub></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">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:92_1_2_1_4_3_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E11:92_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="finseq_1.html#E14:92_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2_1_4_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:92_1_2_1_4_3_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> ) )
 <b>by </b><i><a class="txt" href="finseq_1.html#E7:92_1_2_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E5:92_1_2_1_4_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E3:92_1_2_1_4_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T44" target="_self">Th44</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>16</sub></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">E69</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></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">E70</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E7:92_1_2_1_4_3_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="finseq_1.html#E7:92_1_2_1_4_3_1"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E10:92_1_2_1_4_3_1"/><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E8:92_1_2_1_4_3_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:92_1_2_1_4_3_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_4_3_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="finseq_1.html#E7:92_1_2_1_4_3_1"><i><font color="Green">E69</font></i></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">E73</font></i>: 
( <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> is   <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></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">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <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#D4" target="_self">Def4</a></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">c<sub>9</sub></font> &amp;  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E10:92_1_2_1_4_3_1"><i><font color="Green">E71</font></i></a>, <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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>
 
;<br/>
<a NAME="E4:92_1_2_1_4_3_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:92_1_2_1_4_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_4_3_1_2"><i><font color="Green">E74</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:92_1_2_1_4_3_1_2"/>
<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_4_3_1_2"><i><font color="Green">E75</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">c<sub>18</sub></font> = <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E14:92_1_2_1_4_3_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_4_3_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_4_3_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="finseq_1.html#E12:92_1_2_1_4_3_1"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:92_1_2_1_4_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2_1_4_3_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="finseq_1.html#E7:92_1_2_1_4_3_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="finseq_1.html#E11:92_1_2_1_4_3_1"><i><font color="Green">E72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:92_1_2_1_4_3"/>
<font color="Maroon">c<sub>9</sub></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">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_4_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T57" target="_self">Th57</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:92_1_2_1_4"/>
<font color="Maroon">c<sub>12</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2_1_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:92_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></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">E63</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></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">c<sub>14</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">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">c<sub>14</sub></font>
;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1"><i><font color="Green">E55</font></i></a>, <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">E64</font></i>: 
( <font color="Maroon">c<sub>5</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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></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">c<sub>5</sub></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="txt" href="finseq_1.html#E3:92_1_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T50" target="_self">Th50</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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E65</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></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">E66</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></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">E67</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_2_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_2_1_1"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E6:92_1_2_1_5_2_1_1"/><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1_5_2_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:92_1_2_1_5_2_1_1"/><i><font color="Green">E69</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1_5_2_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_2_1_1_1"/>
<font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:92_1_2_1_5_2_1_1"><i><font color="Green">E68</font></i></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">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>6</sub></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#T56" target="_self">Th56</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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#T35" target="_self">Th35</a></i>
;<br/>
<b>hence </b><a NAME="E4:92_1_2_1_5_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">E70</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_2_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E9:92_1_2_1_5_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59" target="_self">Th59</a></i>;<br/>
<b>hence </b><a NAME="E10:92_1_2_1_5_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="finseq_1.html#E7:92_1_2_1_5_2_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="finseq_1.html#E8:92_1_2_1_5_2_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E66</font></i>: <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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#E3:92_1_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T44" target="_self">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self">Th56</a></i>
;<br/>
<a NAME="E3:92_1_2_1_5_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></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">c<sub>14</sub></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">c<sub>14</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></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">c<sub>14</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>14</sub></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">c<sub>5</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></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">E67</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E3:92_1_2_1_5_2_1_3"/><b>then </b><i><font color="Green">E68</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_1"><i><font color="Green">E65</font></i></a>, <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">c<sub>14</sub></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">c<sub>5</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_5_2_1_3"><i><font color="Green">E67</font></i></a>, <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">c<sub>14</sub></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">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_2_1_3"><i><font color="Green">E68</font></i></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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_5_2_1"><i><font color="Green">E66</font></i></a>, <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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></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">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></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">E65</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></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">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></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">c<sub>14</sub></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">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:92_1_2_1_5_2_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E6:92_1_2_1_5_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_2"><i><font color="Green">E65</font></i></a>, <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">c<sub>15</sub></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">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:92_1_2_1_5_2_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></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_2_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></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">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a></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">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>15</sub></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">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_2_2"><i><font color="Green">E65</font></i></a>, <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">c<sub>15</sub></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">c<sub>5</sub></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">c<sub>6</sub></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#T56" target="_self">Th56</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">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></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">c<sub>6</sub></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_1_2_1_5_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E14:92_1_2_1_5_2_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="finseq_1.html#E5:92_1_2_1_5_2_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="finseq_1.html#E9:92_1_2_1_5_2_2"><i><font color="Green">E67</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:92_1_2_1_5"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></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">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">E65</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></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">E66</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></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">E67</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_3_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_3_1_1"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E6:92_1_2_1_5_3_1_1"/><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1_5_3_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:92_1_2_1_5_3_1_1"/><i><font color="Green">E69</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></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">c<sub>13</sub></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">c<sub>13</sub></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">c<sub>13</sub></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">c<sub>6</sub></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">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1_5_3_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/>

<b>thus </b><a NAME="E2:92_1_2_1_5_3_1_1_1"/>
<font color="Maroon">c<sub>15</sub></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">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:92_1_2_1_5_3_1_1"><i><font color="Green">E68</font></i></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">c<sub>13</sub></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">c<sub>13</sub></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">c<sub>6</sub></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#T56" target="_self">Th56</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></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#T35" target="_self">Th35</a></i>
;<br/>
<b>hence </b><a NAME="E4:92_1_2_1_5_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>13</sub></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">c<sub>13</sub></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">c<sub>6</sub></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">E70</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E17:92_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="finseq_1.html#E3:92_1_2_1_5_3_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E9:92_1_2_1_5_3_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E17:92_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T59" target="_self">Th59</a></i>;<br/>
<b>hence </b><a NAME="E10:92_1_2_1_5_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="finseq_1.html#E17:92_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="finseq_1.html#E7:92_1_2_1_5_3_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="finseq_1.html#E8:92_1_2_1_5_3_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E66</font></i>: <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></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#E4:92_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="finseq_1.html#E17:92_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T44" target="_self">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" target="_self">Th56</a></i>
;<br/>
<a NAME="E3:92_1_2_1_5_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></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">c<sub>13</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></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">c<sub>14</sub></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">c<sub>14</sub></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">c<sub>13</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></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">c<sub>14</sub></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">c<sub>13</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>14</sub></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">c<sub>13</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></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">E67</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E3:92_1_2_1_5_3_1_3"/><b>then </b><i><font color="Green">E68</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_1"><i><font color="Green">E65</font></i></a>, <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">c<sub>14</sub></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">c<sub>13</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_5_3_1_3"><i><font color="Green">E67</font></i></a>, <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">c<sub>14</sub></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">c<sub>13</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E3:92_1_2_1_5_3_1_3"><i><font color="Green">E68</font></i></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">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:92_1_2_1_5_3_1"><i><font color="Green">E66</font></i></a>, <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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></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">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></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">E65</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/>

<a NAME="E2:92_1_2_1_5_3_2"/><b>then </b><i><font color="Green">E66</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E11:92_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="finseq_1.html#E14:92_1_2_1"><i><font color="Green">E61</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">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>11</sub></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">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E14:92_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green">E65</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">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></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">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green">E65</font></i></a>, <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">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E14:92_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:92_1_2_1_5_3_2"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E14:92_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2_1_5_3_2"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:92_1_2_1_5_3_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:92_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2_1_5_3_2"><i><font color="Green">E66</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:92_1_2_1_5"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92_1_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2_1_5"><i><font color="Green">E63</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2_1_5"><i><font color="Green">E64</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:92_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font></span><a href="finseq_1.html#K9" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E17:92_1_2_1"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:92_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="finseq_1.html#S3" target="_self">FINSEQ_1:sch 3</a>(<a class="txt" href="finseq_1.html#E3:92_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="finseq_1.html#E4:92_1_2"><i><font color="Green">E53</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:92_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:92"><i><font color="Green">E49</font></i></a>, <a class="txt" href="finseq_1.html#E1:92_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="finseq_1.html#E2:92_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>


</div>
