<?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>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st ( for <font color="Olive">b<sub>3</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:69"/><i><font color="Green">E27</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="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:69_1"/><i><font color="Green">E28</font></i>: 
(  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>4</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>3</sub></font>,<font color="Maroon">c<sub>4</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>3</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>4</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>3</sub></font>,<font color="Maroon">c<sub>4</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>thus </b><a NAME="E2:69_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>4</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:69_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:69_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:69_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/>
<a NAME="E4:69_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E1:69_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:69_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E1:69_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:69_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E3:69_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E3:69_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E4:69_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:69_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<a NAME="E6:69_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E1:69_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:69_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E3:69_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:69_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E5:69_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:69"/><i><font color="Green">E28</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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:69_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E2:69_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <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">c<sub>3</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:69_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:69_2"/><b>then </b><i><font color="Green">E31</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:69_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:69_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <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">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T65" target="_self">Th65</a></i>;<br/><b>hence </b><a NAME="E3:69_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E2:69_2"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:69_2"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E1:69_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_lang1.html#E4:69_2"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:69_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:69"/><i><font color="Green">E29</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="69_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>



<b>assume </b><a NAME="E1:69_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>] )
 ;<br/>

<a NAME="E2:69_3"/><i><font color="Green">E31</font></i>: 
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>4</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>3</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>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T66" target="_self">Th66</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E3:69_3"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E4:69_3"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:69_3"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:69_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:69_3_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E2:69_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:69_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E3:69_3"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:69_3"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E1:69_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zf_lang1.html#E5:69_3"><i><font color="Green">E33</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:69_3_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:69_3_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E2:69_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:69_3_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E3:69_3"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:69_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E1:69_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zf_lang1.html#E5:69_3"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:69_3"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E7:69_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:69"/><i><font color="Green">E30</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="69_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<b>assume </b><a NAME="E1:69_4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:69_4"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <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>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:69_4"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:69_4_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <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>4</sub></font>,<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>3</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>4</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="zf_lang1.html#E3:69_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zf_lang1.html#T67" target="_self">Th67</a>, <a class="ref" href="zf_lang1.html#T80" target="_self">Th80</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:69_4_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:69_4_1_1"/><b>then </b>
( ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <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>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> ) or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="zf_lang1.html#E1:69_4_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:69_4_1_1"/><b>then </b>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> ) or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="zf_lang1.html#E2:69_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:69_4_1_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_lang1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:69_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:69_4_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E1:69_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zf_lang1.html#E1:69_4_1"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_4"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T80" target="_self">Th80</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:69"/>
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#E1:69"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_lang1.html#E2:69"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zf_lang1.html#E3:69"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_lang1.html#E4:69"><i><font color="Green">E30</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:69"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st ( for <font color="Olive">b<sub>3</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
