<?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>assume </b><a NAME="E1:1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> holds <br/><font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a><br/>

<a NAME="E2:1"/>
<font color="Maroon">E</font> <a href="zf_model.html#R2">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K11">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:1_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:1_1_1_1"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 ;<br/>

<a NAME="E2:1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:1_1_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 2 holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<a NAME="E6:1_1_1_1_1"/>
for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> <a href="zf_lang.html#K2">x.</a> 2 <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:1_1_1_1_1"/><b>then </b>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E8:1_1_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> iff <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_model.html#T19">ZF_MODEL:19</a></i>;<br/>
<a NAME="E9:1_1_1_1_1"/>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E6:1_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 2 holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<a NAME="E7:1_1_1_1"/>
for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> <a href="zf_lang.html#K2">x.</a> 2 <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:1_1_1_1"/><b>then </b>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E9:1_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> iff <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_model.html#T19">ZF_MODEL:19</a></i>;<br/>
<a NAME="E10:1_1_1_1"/>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:1_1"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K11">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T18">ZF_MODEL:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:1"/><b>then </b>
<font color="Maroon">E</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K11">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_model.html#T25">ZF_MODEL:25</a></i>;<br/>
<b>hence </b><a NAME="E4:1"/>
<font color="Maroon">E</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K6">the_axiom_of_extensionality</a> 
 <b>by </b><i><a class="ref" href="zf_model.html#T25">ZF_MODEL:25</a></i>;<br/>


</div>
