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

<b>let </b><font color="Maroon">E</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>2</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>] <b>means </b>for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>  st ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; <font color="Maroon">E</font>,<font color="Olive">f</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:16"/><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> is <a href="zf_lang.html#V7">atomic</a> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  or <font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D15">ZF_LANG:def 15</a><br/>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:16_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E3:16_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 ;<br/>

<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_1"/>
<font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a> 
 ;<br/><a NAME="E2:16_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T53">ZF_LANG:53</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/><a NAME="E3:16_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/><a NAME="E4:16_1_1"/>
(  <a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp;  <a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:16_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:16_1_1"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_2"/>
<font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 ;<br/><a NAME="E2:16_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T54">ZF_LANG:54</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/><a NAME="E3:16_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/><a NAME="E4:16_1_2"/>
(  <a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp;  <a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:16_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:16_1_2"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:16_1"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V4">negative</a> &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K20">the_argument_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_2"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V4">negative</a>
 ;<br/>

<a NAME="E2:16_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#D30">ZF_LANG:def 30</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/>
<b>assume </b><a NAME="E3:16_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>  st ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">x</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#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span> ) &amp; <font color="Maroon">E</font>,<font color="Olive">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E4:16_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:16_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E6:16_2"/><i><font color="Green">E55</font></i>: 
not <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E7:16_2"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font>
 
<b>by </b><i>, , <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E8:16_2"/><b>then </b>
( <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font> )
 
<b>by </b><i>, , , , <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>
<b>hence </b><a NAME="E9:16_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:16"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V5">conjunctive</a> &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Olive">H</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_3"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<a NAME="E2:16_3"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T58">ZF_LANG:58</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E3:16_3"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>  st ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">x</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#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> ) &amp; <font color="Maroon">E</font>,<font color="Olive">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E4:16_3"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>  st ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">x</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#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> ) &amp; <font color="Maroon">E</font>,<font color="Olive">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E5:16_3"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E6:16_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E7:16_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font> &amp; <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</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="16_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:16_3_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:16_3_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:16_3_1"/>
not <font color="Maroon">x</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#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:16_3"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:16_3_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:16_3_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:16_3_2"/>
not <font color="Maroon">x</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#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:16_3"/><b>then </b>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E12:16_3"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:16"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V6">universal</a> &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K24">the_scope_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_4"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<a NAME="E2:16_4"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T62">ZF_LANG:62</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/>
<b>assume </b><a NAME="E3:16_4"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>  st ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">x</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#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span> ) &amp; <font color="Maroon">E</font>,<font color="Olive">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">E</font>,<font color="Olive">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E4:16_4"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:16_4"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/><b>assume </b><a NAME="E1:16_4_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 ;<br/><b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E3:16_4_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span> &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><a NAME="E4:16_4_1"/>
for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/> <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 <b>by </b><i/>;<br/><a NAME="E5:16_4_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">h</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:16_4_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:16_4_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><a NAME="E3:16_4_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:16_4_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:16_4_1_1"/>
not <font color="Maroon">x</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#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:16_4_1"/>
<font color="Maroon">E</font>,<font color="Maroon">j</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:16_4"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:16"/>
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 <b>from </b><i><a class="ref" href="zf_lang.html#S1">ZF_LANG:sch 1</a>(, , , );</i><br/>


</div>
