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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font>;<br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c3">X</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/>

<a NAME="E2:40"/><b>then </b><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c3">X</font> is <a href="zf_fund1.html#V7" title="ZF_FUND1:attr.7">closed_wrt_A7</a>
 
<b>by </b><i><a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></i>;<br/>
<a NAME="E3:40"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c3">X</font> is <a href="zf_fund1.html#V4" title="ZF_FUND1:attr.4">closed_wrt_A4</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">f</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">F</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">Z</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span> ;<br/>
<a NAME="E4:40"/><i><font color="Green" title="E46">A4</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_1"/><i><font color="Green" title="E47">A5</font></i>: 
(  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp;  <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E36"><i><font color="Green" title="E35">Lm13</font></i></a></i>;<br/>
<a NAME="E2:40_1"/><b>then </b><i><font color="Green" title="E48">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T6" target="_self" title="ZF_FUND1:th.6">Th6</a></i>;<br/>
<a NAME="E3:40_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E1:40_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="zf_fund1.html#T6" target="_self" title="ZF_FUND1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:40_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E2:40_1"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="zf_fund1.html#T6" target="_self" title="ZF_FUND1:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:40"/><b>then </b><i><font color="Green" title="E47">A7</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">f'</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E6:40"/><i><font color="Green" title="E48">A8</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<a NAME="E1:40_2"/><i><font color="Green" title="E49">A9</font></i>: 
 <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E36"><i><font color="Green" title="E35">Lm13</font></i></a></i>;<br/>
<a NAME="E2:40_2"/><b>then </b><i><font color="Green" title="E50">A10</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">s</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E4:40_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_2_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><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="40_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">p</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="E1:40_2_1_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 ;<br/>

<a NAME="E2:40_2_1_1"/><b>then </b><i><font color="Green" title="E51">A11</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> )
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> =  <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="zf_fund1.html#E1:40_2"><i><font color="Green" title="E49">A9</font></i></a></i>;<br/>
<a NAME="E4:40_2_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">s</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:40_2_1_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">y'</font>,<font color="Olive" title="b2">x'</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">y'</font>, <font color="Olive" title="b2">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Olive" title="b2">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:40_2_1_1"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c9">p</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="E2:40_2_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c11">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:40_2_1"/><i><font color="Green" title="E51">A12</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">x</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> )
 ;<br/>
<a NAME="E5:40_2_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:40_2_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:40_2_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:40_2_1"><i><font color="Green" title="E51">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:40_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E3:40"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="zf_fund1.html#E2:40_2"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="zf_fund1.html#D9" target="_self" title="ZF_FUND1:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">F'</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E8:40"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F'</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_3"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F'</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><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="40_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">p</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="E1:40_3_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:40_3_1"/><i><font color="Green" title="E49">A13</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x'</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="zf_fund1.html#E4:40"><i><font color="Green" title="E46">A4</font></i></a></i>;<br/>
<a NAME="E5:40_3_1"/>
 <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:40"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E36"><i><font color="Green" title="E35">Lm13</font></i></a></i>;<br/>
<a NAME="E6:40_3_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="classes2.html#T64" title="CLASSES2:th.64">CLASSES2:64</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">y</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="classes2.html#T63" title="CLASSES2:th.63">CLASSES2:63</a></i>;<br/>
<a NAME="E8:40_3_1"/><i><font color="Green" title="E50">A14</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F'</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:40_3_1"><i><font color="Green" title="E49">A13</font></i></a></i>;<br/>
<a NAME="E9:40_3_1"/><i><font color="Green" title="E51">A15</font></i>: 
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:40_3_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c12">y</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:40_3_1"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="zf_fund1.html#E43"><i><font color="Green" title="E41">Lm14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:40_3_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">y'</font></span>)</span> where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F'</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E8:40_3_1"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="zf_fund1.html#E9:40_3_1"><i><font color="Green" title="E51">A15</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c9">p</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="E2:40_3"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F'</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:40_3"/><i><font color="Green" title="E49">A16</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F'</font> )
 ;<br/>
<a NAME="E5:40_3"/><i><font color="Green" title="E50">A17</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:40_3"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:40_3"/><i><font color="Green" title="E51">A18</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c12">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:40_3"><i><font color="Green" title="E49">A16</font></i></a></i>;<br/>
<a NAME="E8:40_3"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c12">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c12">x'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:40_3"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="zf_fund1.html#E5:40_3"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="zf_fund1.html#E7:40_3"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="zf_fund1.html#E43"><i><font color="Green" title="E41">Lm14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:40_3"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E7:40_3"><i><font color="Green" title="E51">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:40"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:40"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E5:40"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="zf_fund1.html#E6:40"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="zf_fund1.html#D12" target="_self" title="ZF_FUND1:def.12">Def12</a></i>;<br/>


</div>
