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

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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>] <b>means </b> <a href="zf_model.html#K2">Free</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:146"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:146_1"/>
(  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="zf_lang1.html#T8" target="_self">Th8</a>, <a class="ref" href="zf_lang1.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E2:146_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:146"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">H1</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6">'not'</a> <font color="Olive">H1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H1</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:146_2"/>
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font>
 ;<br/>

<a NAME="E2:146_2"/><b>then </b>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:146_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H1</font>]
 <b>by </b><i><a class="ref" href="zf_lang1.html#T16" target="_self">Th16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:146"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">H1</font>, <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Olive">H2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H1</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">H2</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>



<b>assume </b><a NAME="E1:146_3"/>
(  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H2</font> )
 ;<br/>

<a NAME="E2:146_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E3:146_3"/><b>then </b>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:146_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font>]
 <b>by </b><i><a class="txt" href="zf_lang1.html#E1:146"><i><font color="Green">E46</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:146"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">H1</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Olive">x</font>,<font color="Olive">H1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H1</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<b>assume </b><a NAME="E1:146_4"/><i><font color="Green">E65</font></i>: 
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font>
 ;<br/>

<a NAME="E2:146_4"/>
( <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> &amp;  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E3:146_4"/><b>then </b><i><font color="Green">E66</font></i>: 
(  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> &amp;  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H1</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="zf_lang1.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E4:146_4"/><b>then </b>
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65" target="_self">Th65</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E5:146_4"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H1</font>]
 <b>by </b><i><a class="ref" href="zf_lang1.html#T66" target="_self">Th66</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:146"/>
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<b>from </b><i><a class="ref" href="zf_lang1.html#S1" target="_self">ZF_LANG1:sch 1</a>(<a class="txt" href="zf_lang1.html#E4:146"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zf_lang1.html#T60" target="_self">Th60</a>, <a class="ref" href="zf_lang1.html#T63" target="_self">Th63</a>, <a class="ref" href="zf_lang1.html#T64" target="_self">Th64</a>);<br/></i>
<b>hence </b><a NAME="E6:146"/>
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H</font>
 ;<br/>


</div>
