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

<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> <a href="hidden.html#R2">in</a>  <a href="zfrefle1.html#K2" target="_self">ZF-axioms</a>  holds <br/><font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">H</font>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">M</font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D1" target="_self">ZFREFLE1:def 1</a><br/>

<a NAME="E3:9"/>
(  <a href="zf_model.html#K7">the_axiom_of_pairs</a>  <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  &amp;  <a href="zf_model.html#K8">the_axiom_of_unions</a>  <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  &amp;  <a href="zf_model.html#K9">the_axiom_of_infinity</a>  <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  &amp;  <a href="zf_model.html#K10">the_axiom_of_power_sets</a>  <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>
<a NAME="E4:9"/><b>then </b>
(  <a href="zf_model.html#K7">the_axiom_of_pairs</a>  <a href="hidden.html#R2">in</a>  <a href="zfrefle1.html#K2" target="_self">ZF-axioms</a>  &amp;  <a href="zf_model.html#K8">the_axiom_of_unions</a>  <a href="hidden.html#R2">in</a>  <a href="zfrefle1.html#K2" target="_self">ZF-axioms</a>  &amp;  <a href="zf_model.html#K9">the_axiom_of_infinity</a>  <a href="hidden.html#R2">in</a>  <a href="zfrefle1.html#K2" target="_self">ZF-axioms</a>  &amp;  <a href="zf_model.html#K10">the_axiom_of_power_sets</a>  <a href="hidden.html#R2">in</a>  <a href="zfrefle1.html#K2" target="_self">ZF-axioms</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:9"/>
( <font color="Maroon">M</font> is <a href="ordinal1.html#V1">epsilon-transitive</a> &amp; <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K7">the_axiom_of_pairs</a>  &amp; <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K8">the_axiom_of_unions</a>  &amp; <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K9">the_axiom_of_infinity</a>  &amp; <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K10">the_axiom_of_power_sets</a>  )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D12">ZF_MODEL:def 12</a><br/>

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

<a NAME="E6:9"/><i><font color="Green">E56</font></i>: 
 <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>
<b>assume </b><a NAME="E7:9"/>
<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>
 ;<br/>

<a NAME="E8:9"/><b>then </b>
 <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="zfrefle1.html#K2" target="_self">ZF-axioms</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:9"/>
<font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


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