<?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="E1:45"/><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>let </b><font color="Maroon">H'</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>



<b>assume </b><a NAME="E2:45"/><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> &amp;  <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>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">E'</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="E4:45"/><i><font color="Green">E61</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><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> <a href="subset_1.html#K7">\</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></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/>
<b>then reconsider </b><font color="Maroon">F1</font> =  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><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> <a href="subset_1.html#K7">\</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></span>)</span>,<font color="Maroon">E'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E6:45"/><i><font color="Green">E62</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><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> <a href="subset_1.html#K7">\</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></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/>
<b>then reconsider </b><font color="Maroon">F2</font> =  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><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> <a href="subset_1.html#K7">\</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></span>)</span>,<font color="Maroon">E'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> ;<br/>
<b>reconsider </b><font color="Maroon">D1</font> =  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">H</font>,<font color="Maroon">E</font>, <font color="Maroon">D2</font> =  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">H'</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/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) </span>}</span> ;<br/>
<a NAME="E9:45"/>
<font color="Maroon">X</font> is <a href="zf_fund1.html#V5" target="_self">closed_wrt_A5</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:45"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E11:45"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H'</font></span>)</span>,<font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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/><b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) </span>}</span> 
 ;<br/><a NAME="E2:45_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> ) </span>}</span>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:45_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="classes2.html#K7">\/</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> )
 ;<br/><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:45_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="classes2.html#K7">\/</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> )
 <b>by </b><i><a class="ref" href="zf_fund1.html#D1" target="_self">Def1</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="E8:45_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</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> <font color="Maroon">H</font>,<font color="Maroon">E</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E10:45_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">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="Maroon">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><a class="txt" href="zf_fund1.html#E1:45"><i><font color="Green">E56</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">e</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:45_1_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">y</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> <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> <a href="subset_1.html#K7">\</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;  <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">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E14:45_1_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</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> <a href="subset_1.html#K7">\</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;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:45"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E15:45_1_1"/><i><font color="Green">E108</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="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></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> <font color="Maroon">H</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<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="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></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> )
 <b>by </b><i/>;<br/><a NAME="E16:45_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="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></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><b>then reconsider </b><font color="Maroon">gg</font> = <span class="p1">(<span class="default"><span class="p2">(<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="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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">e</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="grfunc_1.html#T31">GRFUNC_1:31</a></i>;<br/><a NAME="E18:45_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</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> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><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> <a href="subset_1.html#K7">\</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></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">e</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="zf_fund1.html#E4:45"><i><font color="Green">E61</font></i></a>, <a class="ref" href="grfunc_1.html#T33">GRFUNC_1:33</a></i>;<br/><a NAME="E19:45_1_1"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</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> <a href="subset_1.html#K4">\/</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;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">gg</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/><a NAME="E20:45_1_1"/><b>then </b>
<font color="Maroon">gg</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><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> <a href="subset_1.html#K4">\/</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></span>)</span>,<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>then consider </b><font color="Maroon">ff</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="E22:45_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">ff</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"><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> <a href="subset_1.html#K4">\/</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></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E23:45_1_1"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ff</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> <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:45_1_1_1_1"/><i><font color="Green">E111</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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="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></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="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></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> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:45_1_1_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</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>
 ;<br/><a NAME="E3:45_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">ff</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"><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> <a href="subset_1.html#K4">\/</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></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="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></span>)</span> <a href="relset_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="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></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:45_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E4:45_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="grfunc_1.html#T34">GRFUNC_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:45"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E2:45_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E4:45_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="grfunc_1.html#T34">GRFUNC_1:34</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">ff</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> <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="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:45_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">ff</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:45_1_1_2_1"/><i><font color="Green">E118</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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="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></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="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></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> <font color="Maroon">H'</font></span>)</span> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:45_1_1_2_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</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>
 ;<br/><a NAME="E3:45_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">ff</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"><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> <a href="subset_1.html#K4">\/</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></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="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></span>)</span> <a href="relset_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="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></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:45_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E4:45_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="grfunc_1.html#T35">GRFUNC_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
<b>by </b><i>, <a class="txt" href="zf_fund1.html#E1:45"><i><font color="Green">E56</font></i></a>, <a class="txt" href="zf_fund1.html#E2:45"><i><font color="Green">E57</font></i></a>, , , , <a class="txt" href="zf_fund1.html#E6:45"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E2:45_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E4:45_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="grfunc_1.html#T34">GRFUNC_1:34</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">ff</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">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>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:45_1_1"/><b>then </b>
( <font color="Maroon">ff</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> &amp; <font color="Maroon">ff</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="txt" href="zf_fund1.html#E2:45"><i><font color="Green">E57</font></i></a>, , <a class="txt" href="zf_fund1.html#E10:45"><i><font color="Green">E63</font></i></a>, </i>;<br/><a NAME="E26:45_1_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H'</font></span>)</span>,<font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T5">ZF_MODEL:5</a></i>;<br/><font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">ff</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"><span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K4">\/</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></span>)</span>
<b>by </b><i>, <a class="txt" href="zf_fund1.html#E2:45"><i><font color="Green">E57</font></i></a>, , <a class="txt" href="zf_fund1.html#E6:45"><i><font color="Green">E62</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</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> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T66">ZF_LANG1:66</a></i>
;<br/><b>hence </b><a NAME="E28:45_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"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H'</font></span>)</span>,<font color="Maroon">E</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:45_1_1"><i><font color="Green">E64</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H'</font></span>)</span>,<font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E3:45_1"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H'</font></span>)</span>,<font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_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"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</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="E3:45_1_2"/><i><font color="Green">E121</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> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H'</font></span>)</span></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"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H'</font></span>)</span>,<font color="Maroon">E</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:45_1_2"/><i><font color="Green">E122</font></i>: 
( <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> &amp; <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>, <a class="ref" href="zf_model.html#T5">ZF_MODEL:5</a></i>;<br/>
<a NAME="E5:45_1_2"/><i><font color="Green">E123</font></i>: 
 <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H'</font></span>)</span> <a href="hidden.html#R1">=</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="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T66">ZF_LANG1:66</a></i>;<br/>
<b>then </b><i><font color="Green">E124</font></i>: <font color="Maroon">p</font> = 
<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"><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> <a href="subset_1.html#K4">\/</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></span>)</span>
<b>by </b><i><a class="ref" href="zf_fund1.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
<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"><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> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H'</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="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></span>)</span> <a href="relset_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H'</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>
;<br/>
<a NAME="E7:45_1_2"/><i><font color="Green">E125</font></i>: 
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <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> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<a NAME="E9:45_1_2"/>
<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"><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> <a href="subset_1.html#K7">\</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></span>)</span> is   <a href="funct_2.html#NM1">Function</a> of <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> <a href="subset_1.html#K7">\</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><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E10:45_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<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"><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> <a href="subset_1.html#K7">\</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></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <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"><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> <a href="subset_1.html#K7">\</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></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<a NAME="E12:45_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="classes2.html#K7">\/</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:45"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E13:45_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E128</font></i>: <font color="Maroon">p</font> = 
<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"><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> <a href="subset_1.html#K4">\/</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></span>)</span>
<b>by </b><i><a class="ref" href="zf_fund1.html#D1" target="_self">Def1</a>, <a class="txt" href="zf_fund1.html#E1:45"><i><font color="Green">E56</font></i></a>, </i>
<br/>.= 
<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"><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> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H'</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="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></span>)</span> <a href="relset_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">H'</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>
;<br/>
<a NAME="E15:45_1_2"/><i><font color="Green">E135</font></i>: 
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <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> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<a NAME="E17:45_1_2"/>
<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"><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> <a href="subset_1.html#K7">\</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></span>)</span> is   <a href="funct_2.html#NM1">Function</a> of <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> <a href="subset_1.html#K7">\</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><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E18:45_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<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"><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> <a href="subset_1.html#K7">\</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></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <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"><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> <a href="subset_1.html#K7">\</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></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<a NAME="E20:45_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="classes2.html#K7">\/</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:45"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E21:45_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:45"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E10:45"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:45_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="classes2.html#K7">\/</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:45"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K7">'&amp;'</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>
