<?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>let </b><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c3">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/>





<a NAME="E1:4"/><i><font color="Green" title="E3">A1</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d'</font> where <font color="Olive" title="b1">d'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> : for <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>  st <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d'</font> holds <br/> ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <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> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="zf_colla.html#T1" target="_self" title="ZF_COLLA:th.1">Th1</a></i>;<br/>
<a NAME="E2:4"/><i><font color="Green" title="E4">A2</font></i>: 
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> <font color="Maroon" title="c1">E</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d'</font> where <font color="Olive" title="b1">d'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> : for <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>  st <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d'</font> holds <br/> ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</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> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="zf_colla.html#T1" target="_self" title="ZF_COLLA:th.1">Th1</a></i>;<br/>
<b>thus </b><a NAME="E3:4"/>
( <font color="Maroon" title="c3">d</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">E</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> implies <font color="Maroon" title="c3">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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green" title="E5">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">d</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">E</font> holds <br/><font color="Olive" title="b1">a</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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">d1</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="E1:4_1_1"/>
<font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">d</font>
 ;<br/><a NAME="E2:4_1_1"/><b>then </b>
<font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">d</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:4_1_1"/><b>then </b>
( <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:4_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/><b>hence </b><a NAME="E4:4_1_1"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">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">B</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_1"/>
<font color="Maroon" title="c3">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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_colla.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E4:4"/>
<font color="Maroon" title="c3">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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/>

<a NAME="E5:4"/><b>then </b><i><font color="Green" title="E5">A4</font></i>: 
 ex <font color="Olive" title="b1">e1</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">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp; ( for <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>  st <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e1</font> holds <br/> ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</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> ) ) )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">a</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="E6:4"/><i><font color="Green" title="E6">A5</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">d</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">E</font>
 ;<br/>

<a NAME="E7:4"/><b>then </b><i><font color="Green" title="E7">A6</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c4">a</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#E6:4"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E10:4"/><i><font color="Green" title="E8">A7</font></i>: 
( <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">e</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="c6">B</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E5:4"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="zf_colla.html#E7:4"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">d1</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="E1:4_2"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c7">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">e</font>
 ;<br/><a NAME="E2:4_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="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d'</font> where <font color="Olive" title="b1">d'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> : for <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>  st <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d'</font> holds <br/> ex <font color="Olive" title="b3">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</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">C</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="zf_colla.html#T1" target="_self" title="ZF_COLLA:th.1">Th1</a></i>;<br/><a NAME="E3:4_2"/><b>then </b>
 ex <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 <br/>( <font color="Olive" title="b1">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; ( for <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>  st <font color="Olive" title="b2">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d'</font> holds <br/> ex <font color="Olive" title="b3">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</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">C</font> ) ) )
 <b>by </b><i><a class="txt" href="zf_colla.html#E10:4"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:4_2"/><i><font color="Green" title="E10">A9</font></i>: 
( <font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c7">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="c8">C</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:4_2"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c8">C</font>;<br/><a NAME="E6:4_2"/>
<font color="Maroon" title="c6">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#E10:4"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/><b>hence </b><a NAME="E7:4_2"/>
( <font color="Maroon" title="c9">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="c7">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">C</font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E5:4_2"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:4"/>
<font color="Maroon" title="c4">a</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#E1:4"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>


</div>
