<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">E</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Olive" title="b2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/>for <font color="Olive" title="b3">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Olive" title="b3">H</font>,<font color="Olive" title="b2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/> <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Olive" title="b3">H</font></span>)</span>,<font color="Olive" title="b2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">E</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c2">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Olive" title="b1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>for <font color="Olive" title="b2">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Olive" title="b2">H</font>,<font color="Olive" title="b1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/> <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Olive" title="b2">H</font></span>)</span>,<font color="Olive" title="b1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">E</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/> <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Olive" title="b1">H</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:44"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c2">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> 
 <span class="kw">and </span><br/><a NAME="E2:44"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/> <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Olive" title="b1">H</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">m</font> = <font color="Maroon" title="c3">E</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E48">A2</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:44"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">fs</font> =  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span>;<br/>
<a NAME="E5:44"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
 <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T65" title="ZF_LANG1:th.65">ZF_LANG1:65</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font> iff ( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font> &amp; not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font> ) )</span><br/><div><span class="lab"><font color="Green" title="E51">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:44_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font> &amp; not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font><span class="kw"> such that </span><br/><a NAME="E3:44_1_1"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:44_1_1"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></span>;<br/><span class="kw">thus </span><a NAME="E5:44_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="zf_fund1.html#E16"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font></span><br/><span class="kw">thus </span><a NAME="E6:44_1_1"/>
not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:44_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:44_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font> st <br/>( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></span>;<br/>
<a NAME="E3:44_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="zf_fund1.html#E23"><span class="lab"><font color="Green" title="E22">Lm10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:44_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="zf_model.html#T4" title="ZF_MODEL:th.4">ZF_MODEL:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:44_1_2"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">and </span><br/><a NAME="E2:44_1_2"/><span class="lab"><font color="Green" title="E53">A9</font></span>: 
not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c8">e</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:44_1_2"/><span class="lab"><font color="Green" title="E54">A10</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font>
 <span class="kw">and </span><br/><a NAME="E5:44_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:44_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_2"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font><span class="kw"> such that </span><br/><a NAME="E8:44_1_2"/><span class="lab"><font color="Green" title="E55">A11</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_2"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E4:44_1_2"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="zf_fund1.html#E24"><span class="lab"><font color="Green" title="E23">Lm11</font></span></a></span>;<br/><a NAME="E9:44_1_2"/>
not <font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1_2"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E4:44_1_2"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E8:44_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></span>;<br/><a NAME="E10:44_1_2"/><span class="kw">then </span>
(  <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T65" title="ZF_LANG1:th.65">ZF_LANG1:65</a>, <a class="ref" href="zf_model.html#T4" title="ZF_MODEL:th.4">ZF_MODEL:4</a></span>;<br/><span class="kw">hence </span><a NAME="E11:44_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44_1_2"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E8:44_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:44_1"/>
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font> iff ( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font> &amp; not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A12</font></span>: 
 <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c3">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E8:44"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="zf_fund1.html#T9" target="_self" title="ZF_FUND1:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:44"/>
 <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">H</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E7:44"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="zf_fund1.html#T4" target="_self" title="ZF_FUND1:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
