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

<b>let </b><font color="Maroon">c<sub>1</sub></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>] means  <a href="zf_model.html#K2">Free</a> <font color="Maroon">a<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>;<br/>
<a NAME="E1:70"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:70_1"/>
(  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T63" target="_self">Th63</a>, <a class="ref" href="zf_lang1.html#T64" target="_self">Th64</a></i>;<br/>
<b>hence </b><a NAME="E2:70_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>3</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>3</sub></font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:70"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6">'not'</a> <font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65" target="_self">Th65</a></i>;<br/>
<a NAME="E3:70"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>



<a NAME="E1:70_2"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T66" target="_self">Th66</a></i>;<br/>
<b>hence </b><a NAME="E2:70_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font>] )
 <b>by </b><i><a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:70"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:70_3"/>
 <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T67" target="_self">Th67</a></i>;<br/>
<b>hence </b><a NAME="E2:70_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>] )
 <b>by </b><i><a class="ref" href="finset_1.html#T16">FINSET_1:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:70"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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#E1:70"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zf_lang1.html#E2:70"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_lang1.html#E3:70"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zf_lang1.html#E4:70"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:70"/>
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


</div>
