<?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="E1:21"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">E</font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 ;<br/>

<b>thus </b><a NAME="E2:21"/>
( ( for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H</font> holds <br/><font color="Maroon">E</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Olive">H</font> ) implies for <font color="Olive">F</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">F</font> <a href="zfmodel1.html#R2" target="_self">is_parametrically_definable_in</a> <font color="Maroon">E</font> holds <br/>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">F</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H</font> holds <br/><font color="Maroon">E</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Olive">H</font>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>given </b><font color="Maroon">H</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">f</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><a NAME="E2:21_1"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E3:21_1"/><i><font color="Green">E41</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> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<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> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H</font> <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> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:21_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfmodel1.html#D4" target="_self">ZFMODEL1:def 4</a><br/>

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

<b>assume </b><a NAME="E5:21_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">X</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> ;<br/>
<a NAME="E7:21_1"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E8:21_1"/>
<font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:21"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">F</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">F</font> <a href="zfmodel1.html#R2" target="_self">is_parametrically_definable_in</a> <font color="Maroon">E</font> holds <br/>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">F</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/>

<a NAME="E4:21"/>
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>  st <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H</font> &amp; <font color="Maroon">E</font>,<font color="Olive">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> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<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> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Olive">H</font> <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> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/>for <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> holds  <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Olive">H</font>,<font color="Olive">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:21_2"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E2:21_2"/><i><font color="Green">E45</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> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<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> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H</font> <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> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:21_2"/><i><font color="Green">E48</font></i>: 
 <a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">f</font> <a href="zfmodel1.html#R2" target="_self">is_parametrically_definable_in</a> <font color="Maroon">E</font>
 
<b>by </b><i>, , </i>;<br/>
<b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>;<br/>

<b>thus </b><a NAME="E4:21_2"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:21"/>
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H</font> holds <br/><font color="Maroon">E</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Olive">H</font>
 <b>by </b><i>, </i>;<br/>


</div>
