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

<b>consider </b><font color="Maroon">phi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E2:12_1"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E3:12_1"/><i><font color="Green">E37</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>and </b><br/><a NAME="E4:12_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">A</font>,<span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font></span>)</span>)
 <b>and </b><br/><a NAME="E5:12_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">A</font>,<span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font></span>)</span>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S11">ORDINAL2:sch 11</a>();<br/></i>
<a NAME="E6:12_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:12_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, , </i>;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:12_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font>
 ;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/><a NAME="E3:12_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">A</font>,<font color="Maroon">B</font>)
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E4:12_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1_2"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font>
 ;<br/><a NAME="E2:12_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">y</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><b>assume </b><a NAME="E3:12_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E4:12_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span>)
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E5:12_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">phi</font> = <font color="Maroon">phi</font> as    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, <a class="ref" href="ordinal4.html#D3">ORDINAL4:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">phi</font>
;<br/>

<a NAME="E8:12_1"/>
 <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<b>hence </b><a NAME="E9:12_1"/>
<font color="Maroon">phi</font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i>, , <a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/>

<b>thus </b><a NAME="E10:12_1"/>
for <font color="Olive">a</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">phi</font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Olive">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal4.html#K4">.</a> <font color="Olive">a</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:12_1_2"/>
 <a href="ordinal4.html#K6">succ</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<b>hence </b><a NAME="E2:12_1_2"/>
<font color="Maroon">phi</font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">a</font></span>)</span>)
 <b>by </b><i>, </i>;<br/>


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

<b>let </b><font color="Maroon">a</font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E11:12_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<b>hence </b><a NAME="E12:12_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">a</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  implies <font color="Maroon">phi</font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">a</font></span>)</span>) )
 <b>by </b><i>, </i>;<br/>


</div>
