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

<b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>,  <a href="funct_1.html#NM1">Function</a>, non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ] <b>means </b>(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Olive">d</font> ) );<br/>
<a NAME="E1:10"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Olive">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">A</font></span>)</span>,<font color="Maroon">E</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>





<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> &amp; ( for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">d</font> ) )
 ;<br/>

<a NAME="E2:10_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:10_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>;<br/>

<b>assume </b><a NAME="E4:10_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font>
 ;<br/>

<a NAME="E5:10_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_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> ;<br/>

<b>assume </b><a NAME="E1:10_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:10_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<a NAME="E6:10_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:10_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E8:10_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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

<b>thus </b><a NAME="E2:10_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T161">RELAT_1:161</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
<b>by </b><i>, , , </i>
;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>for <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">f1</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">f2</font>,<font color="Maroon">E</font>] holds <br/><font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font>;<br/>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>]
 ;<br/><b>thus </b><a NAME="E2:10_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f1</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:10_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">f1</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">f2</font>,<font color="Maroon">E</font>] )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_2_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font>
 ;<br/><a NAME="E2:10_2_1_1"/>
 <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/><a NAME="E4:10_2_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:10_2_1_1"/>
<font color="Maroon">f1</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_2_1_1_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_2_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_2_1_1_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:10_2_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:10_2_1_1_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E8:10_2_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E9:10_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>,<font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>,<font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">E</font>] )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:10_2_1_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span> &amp;  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> )
 
<b>by </b><i>, , , </i>;<br/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E12:10_2_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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

<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="E2:10_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:10_2_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:10_2_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:10_2_1_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E9:10_2_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E10:10_2_1_1_1"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>,<font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>,<font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">E</font>] )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:10_2_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span> &amp;  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> )
 
<b>by </b><i>, , , </i>;<br/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E13:10_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:10_2_1_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_2_1"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:10"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">f</font>,<font color="Maroon">E</font>];<br/>
<a NAME="E4:10"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">B</font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:10_3"/>
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> iff  ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d1</font> ) ) ) );<br/>
<a NAME="E2:10_3"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_3_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:10_3_1"/>
 <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:10_3_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_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> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>given </b><font color="Maroon">B1</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E1:10_3_1_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B1</font>,<font color="Maroon">f1</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 ;<br/>

<b>given </b><font color="Maroon">B2</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E2:10_3_1_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B2</font>,<font color="Maroon">f2</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 ;<br/>

<div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_3_1_1_1"/>
<font color="Maroon">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B2</font>
 ;<br/><a NAME="E2:10_3_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B1</font>,<font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B1</font></span>)</span>,<font color="Maroon">E</font>]
 <b>by </b><i>, </i>;<br/><a NAME="E3:10_3_1_1_1"/><b>then </b>
<font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:10_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_3_1_1_2"/>
<font color="Maroon">B2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B1</font>
 ;<br/><a NAME="E2:10_3_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B2</font>,<font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B2</font></span>)</span>,<font color="Maroon">E</font>]
 <b>by </b><i>, </i>;<br/><a NAME="E3:10_3_1_1_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:10_3_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:10_3_1"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1">TARSKI:sch 1</a>();<br/></i>
<b>take </b><font color="Maroon">y</font> = <font color="Maroon">X</font>;<br/>

<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E7:10_3_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/>

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

<b>thus </b><a NAME="E8:10_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies  ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d1</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_3_1_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> &amp;  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:10_3_1_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>,<font color="Maroon">f</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:10_3_1_2"/>
 <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">d1</font>
;<br/>

<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E8:10_3_1_2"/>
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>,<font color="Maroon">f</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> )
 <b>by </b><i>, </i>;<br/>


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

<b>given </b><font color="Maroon">d1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>, <font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E9:10_3_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>,<font color="Maroon">f</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> )
 ;<br/>

<b>thus </b><a NAME="E10:10_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10_3"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_2"><b>proof </b></a><div class="add">

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





<b>assume </b><a NAME="E1:10_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font>
 ;<br/>

<b>given </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">e1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><a NAME="E2:10_3_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff  ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d1</font> ) ) ) )
 ;<br/>

<b>given </b><font color="Maroon">X2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">e2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><a NAME="E3:10_3_2"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> iff  ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e2</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d1</font> ) ) ) )
 ;<br/>

<a NAME="E4:10_3_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_3_2_1"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> holds <br/><font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_3_2_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/>

<a NAME="E2:10_3_2_1_1"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d1</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:10_3_2_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i>, </i>;<br/>


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

<b>let </b><font color="Maroon">v</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="E2:10_3_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E3:10_3_2_1"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e2</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d1</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:10_3_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_3_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:10_3"/><i><font color="Green">E90</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">E</font>] );<br/>
<a NAME="E6:10_3"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">C</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:10_3_3"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">C</font>]
 ;<br/>

<b>assume </b><a NAME="E2:10_3_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:10_3_3"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:10_3_3"/><i><font color="Green">E95</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>;<br/>

<b>assume </b><a NAME="E5:10_3_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font>
 ;<br/>

<a NAME="E6:10_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E8:10_3_3"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d'</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Olive">f</font>,<font color="Maroon">E</font>] &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d1</font> ) ) ) )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E9:10_3_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_3_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="E1:10_3_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">d1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>, <font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:10_3_3_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d'</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font>,<font color="Maroon">h</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">C'</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:10_3_3_1"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">C'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C'</font> )
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E6:10_3_3_1"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C'</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>,<font color="Olive">h</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">C</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:10_3_3_1_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C'</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font>,<font color="Maroon">h</font>,<font color="Maroon">E</font>] &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> )
 ;<br/>

<a NAME="E2:10_3_3_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E3:10_3_3_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:10_3_3_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
( <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp;  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:10_3_3_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font></span>)</span>,<font color="Maroon">E</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:10_3_3_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:10_3_3_1_1"/>
( <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>
<a NAME="E8:10_3_3_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E9:10_3_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <b>by </b><i>, , , , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10_3_3_1"/>
( <font color="Maroon">C'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_3_3_1_2"/>
<font color="Maroon">C'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:10_3_3_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C'</font>,<font color="Maroon">h</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C'</font></span>)</span>,<font color="Maroon">E</font>] &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:10_3_3_1_2"/>
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">C'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C'</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E4:10_3_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:10_3_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:10_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_3_2"><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="E1:10_3_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</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="E3:10_3_3_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:10_3_3_2"/>
 <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:10_3_3_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E8:10_3_3_2"/>
<font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E9:10_3_3_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font>,<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font></span>)</span>,<font color="Maroon">E</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:10_3_3_2"/>
<font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E11:10_3_3_2"/><b>then </b>
 <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:10_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>
<a NAME="E13:10_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E14:10_3_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:10_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10_3"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();<br/></i>
<a NAME="E8:10_3"/>
( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<b>hence </b><a NAME="E9:10_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">f</font>,<font color="Maroon">E</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:10"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();<br/></i>
<b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:10"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:10"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">f</font>,<font color="Maroon">E</font>]
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E10:10"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>;<br/>

<b>thus </b><a NAME="E11:10"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/>


</div>
