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

<b>let </b><font color="Maroon">V</font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="zf_refle.html#NM1">Subclass</a> of <font color="Maroon">V</font>;<br/><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>assume </b><a NAME="E1:44"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">X</font> is <a href="zf_fund1.html#V8" target="_self">closed_wrt_A1-A7</a> &amp; <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<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="E2:44"/><i><font color="Green">E57</font></i>: 
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>set </b><font color="Maroon">fs</font> =  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">E</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E4:44"/><i><font color="Green">E61</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:44"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<font color="Maroon">E</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">E</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<a NAME="E1:44_1"/><i><font color="Green">E62</font></i>: 
 <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65">ZF_LANG1:65</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">E</font>
 ;<br/><b>then consider </b><font color="Maroon">f</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="E3:44_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">E</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E4:44_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<font color="Maroon">E</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E5:44_1_1_1"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">E</font>
 ;<br/>

<a NAME="E2:44_1_1_1_1"/><b>then </b>
 ex <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 <br/>( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:44_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">E</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:44_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zf_model.html#T4">ZF_MODEL:4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<font color="Maroon">E</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> )
 ;<br/><b>then consider </b><font color="Maroon">e</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:44_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">e</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">f</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="E5:44_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="zf_fund1.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E6:44_1_1_2"/><i><font color="Green">E69</font></i>: 
 <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T65">ZF_LANG1:65</a></i>;<br/><a NAME="E7:44_1_1_2"/>
not <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">H</font>,<font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="zf_fund1.html#D1" target="_self">Def1</a>, , </i>;<br/><a NAME="E8:44_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T4">ZF_MODEL:4</a></i>;<br/><b>hence </b><a NAME="E9:44_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="zf_fund1.html#D1" target="_self">Def1</a>, , <a class="txt" href="zf_fund1.html#E1:44"><i><font color="Green">E56</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:44_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">E</font> iff ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<font color="Maroon">E</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">E</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:44_1"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<font color="Maroon">E</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">E</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="E6:44"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , </i>;<br/>


</div>
