<?xml version="1.0"?>
<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/><b>let </b><font color="Maroon">M</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">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font>
 ;<br/>

<b>thus </b><a NAME="E2:16"/>
( <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> implies <font color="Maroon">M</font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">v</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font> holds  <font color="Maroon">M</font>,<font color="Olive">v</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<b>let </b><font color="Maroon">v</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<a NAME="E2:16_1"/>
( <font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/>
<a NAME="E3:16_1"/><b>then </b>
<font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:16_1"/><b>then </b>
( <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:16_1"/><b>then </b>
( <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> or <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T79">ZF_LANG1:79</a></i>;<br/>
<a NAME="E6:16_1"/><b>then </b>
( <font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> or <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T78">ZF_LANG1:78</a></i>;<br/>
<a NAME="E7:16_1"/><b>then </b>
<font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T78">ZF_LANG1:78</a></i>;<br/>
<b>hence </b><a NAME="E8:16_1"/>
<font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T78">ZF_LANG1:78</a></i>;<br/>


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

<b>assume </b><a NAME="E3:16"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">v</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font> holds  <font color="Maroon">M</font>,<font color="Olive">v</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<b>let </b><font color="Maroon">v</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<a NAME="E4:16"/>
<font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1">/</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:16"/>
<font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div>
