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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:57"/>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="afinsq_1.html#K4">&lt;%&gt;</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">G</font> being   <a href="afinsq_1.html#NM1">XFinSequence</a>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="ordinal4.html#K1">^</a> <span class="p1"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><font color="Maroon">x</font></span><a href="afinsq_1.html#K6">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T44">AFINSQ_1:44</a></i>;<br/>
<a NAME="E4:57"/>
 <a href="afinsq_1.html#K1">len</a> <span class="p1"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><font color="Maroon">x</font></span><a href="afinsq_1.html#K6">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T37">AFINSQ_1:37</a></i>;<br/>
<a NAME="E5:57"/><b>then </b><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">G</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">G</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:57"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:57_1"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:57_1"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">G</font> <a href="ordinal1.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a>, <a class="ref" href="stirl2_1.html#T10" target="_self">Th10</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E6:57_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, <a class="ref" href="stirl2_1.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:57_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:57"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T6" target="_self">Th6</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E8:57"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<a NAME="E9:57"/><b>then </b>
( <font color="Maroon">G</font> is   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a>, <a class="ref" href="stirl2_1.html#T8" target="_self">Th8</a>, <a class="ref" href="afinsq_1.html#T40">AFINSQ_1:40</a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:57"/>
 ex <font color="Olive">G</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">D</font> ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="afinsq_1.html#K5">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><font color="Olive">d</font></span><a href="afinsq_1.html#K9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a></i>;<br/>


</div>
