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

<b>let </b><font color="Maroon" title="c1">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> $3,$1 &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> $3,$1 holds <br/>$2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">d</font> ) );<br/>
<a NAME="E1:10"/><i><font color="Green" title="E9">A1</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">E</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">A</font></span>)</span>,<font color="Maroon" title="c1">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" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>





<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c3">B</font> &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c3">B</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">d</font> ) )
 ;<br/>

<a NAME="E2:10_1"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="zf_colla.html#T4" target="_self" title="ZF_COLLA:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:10_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>;<br/>

<b>assume </b><a NAME="E4:10_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E5:10_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">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" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10_1_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:10_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="zf_colla.html#T7" target="_self" title="ZF_COLLA:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">d1</font> = <font color="Maroon" title="c7">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="zf_colla.html#E3:10_1_1_1"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/>
<a NAME="E6:10_1_1_1"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c8">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E4:10_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_1_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self" title="ZF_COLLA:th.6">Th6</a></i>;<br/>
<a NAME="E7:10_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="zf_colla.html#E2:10_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E8:10_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:10_1_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="zf_colla.html#E6:10_1_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">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" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T161" title="RELAT_1:th.161">RELAT_1:161</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> = 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
<b>by </b><i><a class="txt" href="zf_colla.html#E4:10_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="zf_colla.html#E2:10_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="zf_colla.html#E4:10_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="zf_colla.html#E5:10_1"><i><font color="Green" title="E13">A5</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Olive" title="b1">f1</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">E</font>] holds <br/><font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font>;<br/>
<div><i><font color="Green" title="E10">A8</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" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:10_2"/><i><font color="Green" title="E11">A9</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>]
 ;<br/><b>thus </b><a NAME="E2:10_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">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" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E1:10_2_1"/><i><font color="Green" title="E12">A10</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">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" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:10_2_1_1"/><i><font color="Green" title="E13">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>
 ;<br/><a NAME="E2:10_2_1_1"/>
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="ref" href="zf_colla.html#T7" target="_self" title="ZF_COLLA:th.7">Th7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">d</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2_1_1"><i><font color="Green" title="E13">A11</font></i></a></i>;<br/><a NAME="E4:10_2_1_1"/><i><font color="Green" title="E14">A12</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">d</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="zf_colla.html#E1:10_2_1_1"><i><font color="Green" title="E13">A11</font></i></a></i>;<br/><a NAME="E5:10_2_1_1"/>
<font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">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" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">d</font> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">d</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_2_1_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10_2_1_1_1_1"/><i><font color="Green" title="E15">A13</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:10_2_1_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="zf_colla.html#T7" target="_self" title="ZF_COLLA:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">d1</font> = <font color="Maroon" title="c8">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="zf_colla.html#E3:10_2_1_1_1_1"><i><font color="Green" title="E15">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:10_2_1_1_1_1"/><i><font color="Green" title="E16">A14</font></i>: 
( <font color="Maroon" title="c10">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2_1_1"><i><font color="Green" title="E13">A11</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_2_1_1_1_1"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self" title="ZF_COLLA:th.6">Th6</a></i>;<br/>
<a NAME="E8:10_2_1_1_1_1"/><i><font color="Green" title="E17">A15</font></i>: 
<font color="Maroon" title="c10">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E7:10_2_1_1_1_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">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" title="c10">B</font>,<font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span>,<font color="Maroon" title="c1">E</font>] )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="zf_colla.html#E1:10_2_1"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/>
<a NAME="E10:10_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A16</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span> &amp;  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="zf_colla.html#E7:10_2_1_1_1_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="zf_colla.html#E8:10_2_1_1_1_1"><i><font color="Green" title="E17">A15</font></i></a>, <a class="ref" href="zf_colla.html#T4" target="_self" title="ZF_COLLA:th.4">Th4</a></i>;<br/>
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font>
<b>by </b><i><a class="txt" href="zf_colla.html#E7:10_2_1_1_1_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font>
<b>by </b><i><a class="txt" href="zf_colla.html#E7:10_2_1_1_1_1"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="zf_colla.html#E10:10_2_1_1_1_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E12:10_2_1_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_2_1_1_1_1"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:10_2_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:10_2_1_1_1"/><i><font color="Green" title="E15">A17</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:10_2_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="zf_colla.html#T7" target="_self" title="ZF_COLLA:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">d1</font> = <font color="Maroon" title="c8">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="zf_colla.html#E4:10_2_1_1_1"><i><font color="Green" title="E15">A17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:10_2_1_1_1"/><i><font color="Green" title="E16">A18</font></i>: 
( <font color="Maroon" title="c10">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2_1_1"><i><font color="Green" title="E13">A11</font></i></a>, <a class="txt" href="zf_colla.html#E4:10_2_1_1_1"><i><font color="Green" title="E15">A17</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self" title="ZF_COLLA:th.6">Th6</a></i>;<br/>
<a NAME="E9:10_2_1_1_1"/><i><font color="Green" title="E17">A19</font></i>: 
<font color="Maroon" title="c10">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E8:10_2_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">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" title="c10">B</font>,<font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span>,<font color="Maroon" title="c1">E</font>] )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="zf_colla.html#E1:10_2_1"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/>
<a NAME="E11:10_2_1_1_1"/><b>then </b><i><font color="Green" title="E18">A20</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span> &amp;  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="zf_colla.html#E8:10_2_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="zf_colla.html#E9:10_2_1_1_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="ref" href="zf_colla.html#T4" target="_self" title="ZF_COLLA:th.4">Th4</a></i>;<br/>
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font>
<b>by </b><i><a class="txt" href="zf_colla.html#E8:10_2_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font>
<b>by </b><i><a class="txt" href="zf_colla.html#E8:10_2_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="zf_colla.html#E11:10_2_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E13:10_2_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="zf_colla.html#E4:10_2_1_1_1"><i><font color="Green" title="E15">A17</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">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" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E4:10_2_1_1"><i><font color="Green" title="E14">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_2_1"/>
<font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_2_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:10"/><i><font color="Green" title="E11">A21</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="zf_colla.html#E2:10"><i><font color="Green" title="E10">A8</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b> ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">E</font>];<br/>
<a NAME="E4:10"/><i><font color="Green" title="E12">A22</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">B</font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">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" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:10_3"/>
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/> ex <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">E</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> iff  ex <font color="Olive" title="b4">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b5">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b6">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">d</font> &amp; <font color="Olive" title="b5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b5">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b5">B</font>,<font color="Olive" title="b6">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">d1</font> ) ) ) );<br/>
<a NAME="E2:10_3"/><i><font color="Green" title="E13">A23</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_3_1"/><i><font color="Green" title="E14">A24</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:10_3_1"/>
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="zf_colla.html#T7" target="_self" title="ZF_COLLA:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">d</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_3_1"><i><font color="Green" title="E14">A24</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">E</font>] &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E4:10_3_1"/><i><font color="Green" title="E15">A25</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>given </b><font color="Maroon" title="c8">B1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>, <font color="Maroon" title="c9">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E1:10_3_1_1"/><i><font color="Green" title="E16">A26</font></i>: 
( <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c8">B1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">B1</font>,<font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c10">B2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>, <font color="Maroon" title="c11">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E2:10_3_1_1"/><i><font color="Green" title="E17">A27</font></i>: 
( <font color="Maroon" title="c10">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">B2</font>,<font color="Maroon" title="c11">f2</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 ;<br/>

<div><i><font color="Green" title="E18">A28</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" title="c8">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">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" title="c8">B1</font>,<font color="Maroon" title="c11">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c8">B1</font></span>)</span>,<font color="Maroon" title="c1">E</font>]
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="zf_colla.html#E2:10_3_1_1"><i><font color="Green" title="E17">A27</font></i></a></i>;<br/><a NAME="E3:10_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c8">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f1</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:10"><i><font color="Green" title="E11">A21</font></i></a>, <a class="txt" href="zf_colla.html#E1:10_3_1_1"><i><font color="Green" title="E16">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E4:10_3_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_3_1_1"><i><font color="Green" title="E16">A26</font></i></a>, <a class="txt" href="zf_colla.html#E2:10_3_1_1"><i><font color="Green" title="E17">A27</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">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" title="c10">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">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" title="c10">B2</font>,<font color="Maroon" title="c9">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B2</font></span>)</span>,<font color="Maroon" title="c1">E</font>]
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="zf_colla.html#E1:10_3_1_1"><i><font color="Green" title="E16">A26</font></i></a></i>;<br/><a NAME="E3:10_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:10"><i><font color="Green" title="E11">A21</font></i></a>, <a class="txt" href="zf_colla.html#E2:10_3_1_1"><i><font color="Green" title="E17">A27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:10_3_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:10_3_1_1"><i><font color="Green" title="E16">A26</font></i></a>, <a class="txt" href="zf_colla.html#E2:10_3_1_1"><i><font color="Green" title="E17">A27</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_3_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:10_3_1_1"><i><font color="Green" title="E18">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:10_3_1"/><i><font color="Green" title="E16">A29</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1" title="TARSKI:sch.1">TARSKI:sch 1</a>(<a class="txt" href="zf_colla.html#E4:10_3_1"><i><font color="Green" title="E15">A25</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">X</font>;<br/>

<b>take </b>
<font color="Maroon" title="c5">X</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c4">d</font>
;<br/>

<b>thus </b><a NAME="E7:10_3_1"/>
( <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E8:10_3_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> implies  ex <font color="Olive" title="b1">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b2">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10_3_1_2"/><i><font color="Green" title="E17">A30</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> ) )
 <b>by </b><i><a class="txt" href="zf_colla.html#E6:10_3_1"><i><font color="Green" title="E16">A29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>, <font color="Maroon" title="c10">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:10_3_1_2"/><i><font color="Green" title="E18">A31</font></i>: 
( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c9">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:10_3_1_2"><i><font color="Green" title="E17">A30</font></i></a></i>;<br/>
<a NAME="E6:10_3_1_2"/>
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="zf_colla.html#T7" target="_self" title="ZF_COLLA:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">d1</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="zf_colla.html#E5:10_3_1_2"><i><font color="Green" title="E18">A31</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">d1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">B</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c10">f</font>
;<br/>

<b>thus </b><a NAME="E8:10_3_1_2"/>
( <font color="Maroon" title="c11">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c11">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c9">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">d1</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:10_3_1_2"><i><font color="Green" title="E17">A30</font></i></a>, <a class="txt" href="zf_colla.html#E5:10_3_1_2"><i><font color="Green" title="E18">A31</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c8">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c9">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>, <font color="Maroon" title="c10">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E9:10_3_1"/><i><font color="Green" title="E17">A32</font></i>: 
( <font color="Maroon" title="c8">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c9">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d1</font> )
 ;<br/>

<b>thus </b><a NAME="E10:10_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E6:10_3_1"><i><font color="Green" title="E16">A29</font></i></a>, <a class="txt" href="zf_colla.html#E9:10_3_1"><i><font color="Green" title="E17">A32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10_3"/><i><font color="Green" title="E14">A33</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:10_3_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c6">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c7">e1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><b> such that </b><a NAME="E2:10_3_2"/><i><font color="Green" title="E15">A34</font></i>: 
( <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c7">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X1</font> iff  ex <font color="Olive" title="b2">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b4">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">e1</font> &amp; <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b3">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">d1</font> ) ) ) )
 ;<br/>

<b>given </b><font color="Maroon" title="c8">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">e2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><b> such that </b><a NAME="E3:10_3_2"/><i><font color="Green" title="E16">A35</font></i>: 
( <font color="Maroon" title="c8">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c9">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X2</font> iff  ex <font color="Olive" title="b2">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b4">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> &amp; <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b3">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">d1</font> ) ) ) )
 ;<br/>

<a NAME="E4:10_3_2"/>
<font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X1</font> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c10">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_3_2_1_1"/>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X1</font>
 ;<br/>

<a NAME="E2:10_3_2_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">e1</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b2">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d1</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E2:10_3_2"><i><font color="Green" title="E15">A34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:10_3_2_1_1"/>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X2</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:10_3_2"><i><font color="Green" title="E15">A34</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_2"><i><font color="Green" title="E16">A35</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c10">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:10_3_2_1"/>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X2</font>
 ;<br/>

<a NAME="E3:10_3_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b2">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d1</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E3:10_3_2"><i><font color="Green" title="E16">A35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:10_3_2_1"/>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X1</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:10_3_2"><i><font color="Green" title="E15">A34</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_2"><i><font color="Green" title="E16">A35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_3_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:10_3_2"><i><font color="Green" title="E15">A34</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_2"><i><font color="Green" title="E16">A35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:10_3"/><i><font color="Green" title="E15">A36</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="zf_colla.html#E3:10_3"><i><font color="Green" title="E14">A33</font></i></a>, <a class="txt" href="zf_colla.html#E2:10_3"><i><font color="Green" title="E13">A23</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( $1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,$1</span>)</span>,<font color="Maroon" title="c1">E</font>] );<br/>
<a NAME="E6:10_3"/><i><font color="Green" title="E16">A37</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">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" title="c4">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:10_3_3"/><i><font color="Green" title="E17">A38</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">C</font>]
 ;<br/>

<b>assume </b><a NAME="E2:10_3_3"/><i><font color="Green" title="E18">A39</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E3:10_3_3"/><b>then </b><i><font color="Green" title="E19">A40</font></i>: 
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="zf_colla.html#T4" target="_self" title="ZF_COLLA:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E4:10_3_3"/><i><font color="Green" title="E20">A41</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E5:10_3"><i><font color="Green" title="E15">A36</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>;<br/>

<b>assume </b><a NAME="E5:10_3_3"/><i><font color="Green" title="E21">A42</font></i>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>
 ;<br/>

<a NAME="E6:10_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c7">d'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E8:10_3_3"/><i><font color="Green" title="E22">A43</font></i>: 
( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c7">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> iff  ex <font color="Olive" title="b2">d1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b4">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">d'</font> &amp; <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b3">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">d1</font> ) ) ) )
 <b>by </b><i><a class="txt" href="zf_colla.html#E5:10_3"><i><font color="Green" title="E15">A36</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_3"><i><font color="Green" title="E19">A40</font></i></a>, <a class="txt" href="zf_colla.html#E5:10_3_3"><i><font color="Green" title="E21">A42</font></i></a></i>;<br/>
<a NAME="E9:10_3_3"/><i><font color="Green" title="E23">A44</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">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" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:10_3_3_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c10">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>, <font color="Maroon" title="c11">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:10_3_3_1"/><i><font color="Green" title="E24">A45</font></i>: 
( <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">d'</font> &amp; <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">C</font>,<font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E8:10_3_3"><i><font color="Green" title="E22">A43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">C'</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:10_3_3_1"/><i><font color="Green" title="E25">A46</font></i>: 
( <font color="Maroon" title="c12">C'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c12">C'</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E5:10_3_3"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="zf_colla.html#E8:10_3_3"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_3_1"><i><font color="Green" title="E24">A45</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self" title="ZF_COLLA:th.6">Th6</a></i>;<br/>
<a NAME="E6:10_3_3_1"/><i><font color="Green" title="E26">A47</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c12">C'</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">h</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> holds <br/><font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">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" title="c13">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c14">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E1:10_3_3_1_1"/><i><font color="Green" title="E27">A48</font></i>: 
( <font color="Maroon" title="c13">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c12">C'</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">C</font>,<font color="Maroon" title="c14">h</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">C</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> )
 ;<br/>

<a NAME="E2:10_3_3_1_1"/><b>then </b><i><font color="Green" title="E28">A49</font></i>: 
<font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E5:10_3_3_1"><i><font color="Green" title="E25">A46</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E3:10_3_3_1_1"/><b>then </b>
<font color="Maroon" title="c13">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:10_3_3_1_1"/><b>then </b><i><font color="Green" title="E29">A50</font></i>: 
( <font color="Maroon" title="c13">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E2:10_3_3"><i><font color="Green" title="E18">A39</font></i></a>, <a class="ref" href="zf_colla.html#T4" target="_self" title="ZF_COLLA:th.4">Th4</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="c13">C</font>,<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">C</font></span>)</span>,<font color="Maroon" title="c1">E</font>]
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_3_3"><i><font color="Green" title="E17">A38</font></i></a>, <a class="txt" href="zf_colla.html#E2:10_3_3_1_1"><i><font color="Green" title="E28">A49</font></i></a></i>;<br/>
<a NAME="E6:10_3_3_1_1"/><b>then </b><i><font color="Green" title="E30">A51</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">h</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E3:10"><i><font color="Green" title="E11">A21</font></i></a>, <a class="txt" href="zf_colla.html#E1:10_3_3_1_1"><i><font color="Green" title="E27">A48</font></i></a></i>;<br/>
<a NAME="E7:10_3_3_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">C</font></span>)</span> &amp; <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_3_3_1_1"><i><font color="Green" title="E27">A48</font></i></a>, <a class="txt" href="zf_colla.html#E4:10_3_3_1_1"><i><font color="Green" title="E29">A50</font></i></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></i>;<br/>
<a NAME="E8:10_3_3_1_1"/><b>then </b>
<font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_3_3_1_1"><i><font color="Green" title="E27">A48</font></i></a>, <a class="txt" href="zf_colla.html#E6:10_3_3_1_1"><i><font color="Green" title="E30">A51</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E9:10_3_3_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E4:10_3_3"><i><font color="Green" title="E20">A41</font></i></a>, <a class="txt" href="zf_colla.html#E8:10_3_3"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_3_1"><i><font color="Green" title="E24">A45</font></i></a>, <a class="txt" href="zf_colla.html#E1:10_3_3_1_1"><i><font color="Green" title="E27">A48</font></i></a>, <a class="txt" href="zf_colla.html#E4:10_3_3_1_1"><i><font color="Green" title="E29">A50</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10_3_3_1"/>
( <font color="Maroon" title="c12">C'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">C</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">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" title="c12">C'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">C</font>
 ;<br/>

<a NAME="E2:10_3_3_1_2"/><b>then </b><i><font color="Green" title="E27">A52</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">C'</font>,<font color="Maroon" title="c11">h</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c12">C'</font></span>)</span>,<font color="Maroon" title="c1">E</font>] &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">C</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_3_1"><i><font color="Green" title="E24">A45</font></i></a></i>;<br/>
<a NAME="E3:10_3_3_1_2"/>
( <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c12">C'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> &amp; <font color="Maroon" title="c12">C'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c12">C'</font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E5:10_3_3_1"><i><font color="Green" title="E25">A46</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E4:10_3_3_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:10_3_3_1"><i><font color="Green" title="E24">A45</font></i></a>, <a class="txt" href="zf_colla.html#E5:10_3_3_1"><i><font color="Green" title="E25">A46</font></i></a>, <a class="txt" href="zf_colla.html#E6:10_3_3_1"><i><font color="Green" title="E26">A47</font></i></a>, <a class="txt" href="zf_colla.html#E2:10_3_3_1_2"><i><font color="Green" title="E27">A52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:10_3_3_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:10_3_3_1"><i><font color="Green" title="E24">A45</font></i></a>, <a class="txt" href="zf_colla.html#E6:10_3_3_1"><i><font color="Green" title="E26">A47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:10_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">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" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:10_3_3_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10_3_3_2"/><i><font color="Green" title="E24">A53</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:10_3_3_2"/>
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="zf_colla.html#T7" target="_self" title="ZF_COLLA:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">d1</font> = <font color="Maroon" title="c9">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="zf_colla.html#E4:10_3_3"><i><font color="Green" title="E20">A41</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_3_2"><i><font color="Green" title="E24">A53</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:10_3_3_2"/><i><font color="Green" title="E25">A54</font></i>: 
( <font color="Maroon" title="c11">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c10">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c11">C</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E5:10_3_3"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_3_2"><i><font color="Green" title="E24">A53</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self" title="ZF_COLLA:th.6">Th6</a></i>;<br/>
<a NAME="E8:10_3_3_2"/>
<font color="Maroon" title="c11">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E2:10_3_3"><i><font color="Green" title="E18">A39</font></i></a>, <a class="txt" href="zf_colla.html#E7:10_3_3_2"><i><font color="Green" title="E25">A54</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E9:10_3_3_2"/><b>then </b><i><font color="Green" title="E26">A55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">C</font>,<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c11">C</font></span>)</span>,<font color="Maroon" title="c1">E</font>]
 
<b>by </b><i><a class="txt" href="zf_colla.html#E1:10_3_3"><i><font color="Green" title="E17">A38</font></i></a>, <a class="txt" href="zf_colla.html#E7:10_3_3_2"><i><font color="Green" title="E25">A54</font></i></a></i>;<br/>
<a NAME="E10:10_3_3_2"/>
<font color="Maroon" title="c11">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E7:10_3_3_2"><i><font color="Green" title="E25">A54</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E11:10_3_3_2"/><b>then </b>
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c11">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="zf_colla.html#T4" target="_self" title="ZF_COLLA:th.4">Th4</a></i>;<br/>
<a NAME="E12:10_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c11">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c11">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">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" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c11">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E7:10_3_3_2"><i><font color="Green" title="E25">A54</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E14:10_3_3_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:10_3_3"><i><font color="Green" title="E18">A39</font></i></a>, <a class="txt" href="zf_colla.html#E8:10_3_3"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="zf_colla.html#E3:10_3_3_2"><i><font color="Green" title="E24">A53</font></i></a>, <a class="txt" href="zf_colla.html#E7:10_3_3_2"><i><font color="Green" title="E25">A54</font></i></a>, <a class="txt" href="zf_colla.html#E9:10_3_3_2"><i><font color="Green" title="E26">A55</font></i></a></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" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E8:10_3_3"><i><font color="Green" title="E22">A43</font></i></a>, <a class="txt" href="zf_colla.html#E9:10_3_3"><i><font color="Green" title="E23">A44</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10_3"/><i><font color="Green" title="E17">A56</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">B</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="zf_colla.html#E6:10_3"><i><font color="Green" title="E16">A37</font></i></a>);<br/></i>
<a NAME="E8:10_3"/>
( <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">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" title="c2">A</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">E</font>]
 <b>by </b><i><a class="txt" href="zf_colla.html#E5:10_3"><i><font color="Green" title="E15">A36</font></i></a>, <a class="txt" href="zf_colla.html#E7:10_3"><i><font color="Green" title="E17">A56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:10"/><i><font color="Green" title="E13">A57</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="zf_colla.html#E4:10"><i><font color="Green" title="E12">A22</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c2">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:10"/><i><font color="Green" title="E14">A58</font></i>: 
<font color="Maroon" title="c1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="zf_colla.html#T8" target="_self" title="ZF_COLLA:th.8">Th8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:10"/><i><font color="Green" title="E15">A59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">E</font>]
 <b>by </b><i><a class="txt" href="zf_colla.html#E5:10"><i><font color="Green" title="E13">A57</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/>

<b>thus </b><a NAME="E10:10"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E7:10"><i><font color="Green" title="E14">A58</font></i></a>, <a class="txt" href="zf_colla.html#E9:10"><i><font color="Green" title="E15">A59</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c4">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>;<br/>

<b>thus </b><a NAME="E11:10"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E7:10"><i><font color="Green" title="E14">A58</font></i></a>, <a class="txt" href="zf_colla.html#E9:10"><i><font color="Green" title="E15">A59</font></i></a></i>;<br/>


</div>
