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

<b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">A</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">A</font> &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> holds <br/><font color="Olive">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font> ) );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:6"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>assume </b><a NAME="E3:6"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:6_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> ) ) )
 <b>by </b><i>, <a class="ref" href="tarski.html#T7">TARSKI:7</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i>, </i>;<br/><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">E</font>
 ;<br/><b>assume </b><a NAME="E2:6_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/><b>then consider </b><font color="Maroon">A1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:6_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A1</font> &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> holds <br/><font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font> ) )
 ;<br/><b>assume </b><a NAME="E5:6_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 ;<br/><b>then consider </b><font color="Maroon">A2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:6_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A2</font> &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> holds <br/><font color="Maroon">A2</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font> ) )
 ;<br/><a NAME="E8:6_1_1"/>
( <font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:6_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b><font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">a<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:6_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">E</font>
 ;<br/><a NAME="E2:6_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:6_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E4:6_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">A</font>]
 <b>by </b><i>, </i>;<br/><a NAME="E5:6_1_2"/>
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">A</font>] &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Olive">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i><b>hence </b><a NAME="E6:6_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:6_1"/><i><font color="Green">E56</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">E</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">E</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i><a NAME="E9:6_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> holds <br/><font color="Olive">y</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:6_1_3"/>
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">A</font> &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> holds <br/><font color="Olive">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:6_1_3"/>
<font color="Maroon">y</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E11:6_1"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T36">ORDINAL1:36</a></i>;<br/><a NAME="E12:6_1"/>
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> holds <br/> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_1_4"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:6_1_4"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:6_1_4"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">C</font> holds <br/><font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font> ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<a NAME="E5:6_1_4"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_1_4"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E13:6_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d'</font> where <font color="Olive">d'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d'</font> holds <br/> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">B</font> ) </span>}</span> 
 ;<br/><a NAME="E14:6_1"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Maroon">A</font> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">E</font>,<font color="Olive">A</font> ) )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E15:6_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E5:6"/><b>then </b>
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<b>hence </b><a NAME="E6:6"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div>
