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

<b>let </b><font color="Maroon">M1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:12"/>
( <font color="Maroon">M1</font> <a href="zfrefle1.html#R2" target="_self">&lt;==&gt;</a> <font color="Maroon">M2</font> implies for <font color="Olive">F</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="zf_lang.html#K9">WFF</a>  holds <br/> ( <font color="Maroon">M1</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Olive">F</font> iff <font color="Maroon">M2</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Olive">F</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st  <a href="zf_model.html#K2">Free</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>( <font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">H</font> iff <font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">H</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D2" target="_self">ZFREFLE1:def 2</a><br/>

<b>let </b><font color="Maroon">F</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="zf_lang.html#K9">WFF</a> ;<br/>

<b>thus </b><a NAME="E2:12_1"/>
( <font color="Maroon">M1</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Maroon">F</font> implies <font color="Maroon">M2</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Maroon">F</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1_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 <font color="Olive">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">H</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D1" target="_self">ZFREFLE1:def 1</a><br/>

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D1" target="_self">ZFREFLE1:def 1</a><br/>

<b>consider </b><font color="Maroon">S</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:12_1_1"/><i><font color="Green">E56</font></i>: 
(  <a href="zf_model.html#K2">Free</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">M</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">S</font> iff <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> ) ) )
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:12_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E5:12_1_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:12_1_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:12_1_1"/><b>then </b>
<font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:12_1_1"/>
<font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:12_1"/><i><font color="Green">E57</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> <font color="Maroon">F</font> holds <br/><font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">H</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D1" target="_self">ZFREFLE1:def 1</a><br/>

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D1" target="_self">ZFREFLE1:def 1</a><br/>

<b>consider </b><font color="Maroon">S</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E5:12_1"/><i><font color="Green">E58</font></i>: 
(  <a href="zf_model.html#K2">Free</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">M</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">S</font> iff <font color="Olive">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> ) ) )
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E6:12_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E7:12_1"/><b>then </b>
<font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:12_1"/><b>then </b>
<font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:12_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:12_1"/>
<font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:12"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">F</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="zf_lang.html#K9">WFF</a>  holds <br/> ( <font color="Maroon">M1</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Olive">F</font> iff <font color="Maroon">M2</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Olive">F</font> )
 ;<br/>

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D2" target="_self">ZFREFLE1:def 2</a><br/>

<b>assume </b><a NAME="E3:12"/>
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E4:12"/>
<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>then reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="zf_lang.html#K9">WFF</a>  <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>thus </b><a NAME="E6:12"/>
( <font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> implies <font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:12_2"/><b>then </b>
for <font color="Olive">S</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:12_2"/><b>then </b>
<font color="Maroon">M1</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:12_2"/><b>then </b>
( <font color="Maroon">M2</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:12_2"/>
<font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E7:12"/>
<font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E8:12"/><b>then </b>
for <font color="Olive">S</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">M2</font> <a href="zf_model.html#R2">|=</a> <font color="Olive">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:12"/><b>then </b>
<font color="Maroon">M2</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:12"/><b>then </b>
( <font color="Maroon">M1</font> <a href="zfrefle1.html#R1" target="_self">|=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:12"/>
<font color="Maroon">M1</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


</div>
