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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/><b>let </b><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>;<br/><b>let </b><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> ;<br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E48">A1</font></i>: 
( <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> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">H</font>, <font color="Maroon" title="c5">H'</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>



<b>assume </b><a NAME="E2:45"/><i><font color="Green" title="E49">A2</font></i>: 
(  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c4">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> &amp;  <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> )
 ;<br/>

<b>set </b><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="c4">H</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">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/>
<b>reconsider </b><font color="Maroon" title="c8">E'</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> <b>by </b><i><a class="txt" href="zf_fund1.html#E1:45"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/>
<a NAME="E4:45"/><i><font color="Green" title="E50">A3</font></i>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c4">H</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c8">E'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:45"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T9" target="_self" title="ZF_FUND1:th.9">Th9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">F1</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c4">H</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c8">E'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E6:45"/><i><font color="Green" title="E51">A4</font></i>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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></span>)</span>,<font color="Maroon" title="c8">E'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:45"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T9" target="_self" title="ZF_FUND1:th.9">Th9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">F2</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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></span>)</span>,<font color="Maroon" title="c8">E'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">D1</font> =  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <font color="Maroon" title="c4">H</font>,<font color="Maroon" title="c3">E</font>, <font color="Maroon" title="c12">D2</font> =  <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> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="zf_fund1.html#E2:45"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c14">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> ) </span>}</span> ;<br/>
<a NAME="E9:45"/>
<font color="Maroon" title="c2">X</font> is <a href="zf_fund1.html#V5" title="ZF_FUND1:attr.5">closed_wrt_A5</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:45"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></i>;<br/>
<a NAME="E10:45"/><b>then </b><i><font color="Green" title="E52">A5</font></i>: 
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:45"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E4:45"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="zf_fund1.html#E6:45"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="zf_fund1.html#D10" target="_self" title="ZF_FUND1:def.10">Def10</a></i>;<br/>
<a NAME="E11:45"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span>,<font color="Maroon" title="c3">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" title="c15">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> ) </span>}</span> 
 ;<br/><a NAME="E2:45_1_1"/><b>then </b><i><font color="Green" title="E53">A6</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> ) </span>}</span>  &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:45_1_1"/><i><font color="Green" title="E54">A7</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c18">x'</font>, <font color="Maroon" title="c19">y'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:45_1_1"/><i><font color="Green" title="E55">A8</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x'</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c19">y'</font> &amp; <font color="Maroon" title="c18">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Maroon" title="c19">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:45_1_1"><i><font color="Green" title="E53">A6</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">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><b> such that </b><br/><a NAME="E8:45_1_1"/><i><font color="Green" title="E56">A9</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">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="c4">H</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c20">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="c4">H</font>,<font color="Maroon" title="c3">E</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:45_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">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><b> such that </b><br/><a NAME="E10:45_1_1"/><i><font color="Green" title="E57">A10</font></i>: 
( <font color="Maroon" title="c18">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E6:45_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">e</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:45_1_1"/><i><font color="Green" title="E58">A11</font></i>: 
( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">e</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c4">H</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:45_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E14:45_1_1"/><i><font color="Green" title="E59">A12</font></i>: 
( <font color="Maroon" title="c19">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c23">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E6:45_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E15:45_1_1"/><i><font color="Green" title="E60">A13</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">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="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="c4">H</font></span>)</span></span>)</span></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> <font color="Maroon" title="c4">H</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">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="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="c4">H</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E16"><i><font color="Green" title="E15">Lm3</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">gg</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">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="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="c4">H</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c22">e</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="zf_fund1.html#E12:45_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="grfunc_1.html#T31" title="GRFUNC_1:th.31">GRFUNC_1:31</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><a NAME="E17:45_1_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c4">H</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">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="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">e</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E12:45_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="zf_fund1.html#E15:45_1_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="grfunc_1.html#T33" title="GRFUNC_1:th.33">GRFUNC_1:33</a></i>;<br/><a NAME="E18:45_1_1"/><b>then </b>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">gg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E12:45_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/><a NAME="E19:45_1_1"/><b>then </b>
<font color="Maroon" title="c24">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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></span>)</span>,<font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">ff</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><b> such that </b><br/><a NAME="E21:45_1_1"/><i><font color="Green" title="E61">A14</font></i>: 
<font color="Maroon" title="c24">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">ff</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"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E24"><i><font color="Green" title="E23">Lm11</font></i></a></i>;<br/><a NAME="E22:45_1_1"/><i><font color="Green" title="E62">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">ff</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="c4">H</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">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="c4">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" title="E63">A16</font></i>: 
(  <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="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="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="c4">H</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">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="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="c4">H</font></span>)</span></span>)</span></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> <font color="Maroon" title="c4">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E16"><i><font color="Green" title="E15">Lm3</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c26">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:45_1_1_1_1"/><i><font color="Green" title="E64">A17</font></i>: 
<font color="Maroon" title="c26">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">H</font></span>)</span>
 ;<br/><a NAME="E3:45_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c25">ff</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"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="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="c4">H</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="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></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T107" title="RELAT_1:th.107">RELAT_1:107</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="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="c4">H</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">q</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:45_1_1_1_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="txt" href="zf_fund1.html#E2:45_1_1_1_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="grfunc_1.html#T34" title="GRFUNC_1:th.34">GRFUNC_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">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="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="c4">H</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">q</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E21:45_1_1"><i><font color="Green" title="E61">A14</font></i></a>, <a class="txt" href="zf_fund1.html#E1:45_1_1_1_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="txt" href="zf_fund1.html#E2:45_1_1_1_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="grfunc_1.html#T34" title="GRFUNC_1:th.34">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" title="c25">ff</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="c4">H</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">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="c4">H</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:45_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c25">ff</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> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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>
 <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" title="E63">A18</font></i>: 
(  <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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="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></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">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="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></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> <font color="Maroon" title="c5">H'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E16"><i><font color="Green" title="E15">Lm3</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c26">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:45_1_1_2_1"/><i><font color="Green" title="E64">A19</font></i>: 
<font color="Maroon" title="c26">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 ;<br/><a NAME="E3:45_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c25">ff</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"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="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="c4">H</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="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></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T107" title="RELAT_1:th.107">RELAT_1:107</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">ff</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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">q</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:45_1_1_2_1"><i><font color="Green" title="E63">A18</font></i></a>, <a class="txt" href="zf_fund1.html#E2:45_1_1_2_1"><i><font color="Green" title="E64">A19</font></i></a>, <a class="ref" href="grfunc_1.html#T35" title="GRFUNC_1:th.35">GRFUNC_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">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="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></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">q</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:45_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="zf_fund1.html#E6:45_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="zf_fund1.html#E8:45_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="zf_fund1.html#E10:45_1_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="zf_fund1.html#E12:45_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="zf_fund1.html#E14:45_1_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="txt" href="zf_fund1.html#E21:45_1_1"><i><font color="Green" title="E61">A14</font></i></a>, <a class="txt" href="zf_fund1.html#E1:45_1_1_2_1"><i><font color="Green" title="E63">A18</font></i></a>, <a class="txt" href="zf_fund1.html#E2:45_1_1_2_1"><i><font color="Green" title="E64">A19</font></i></a>, <a class="ref" href="grfunc_1.html#T34" title="GRFUNC_1:th.34">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" title="c25">ff</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> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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>
 <b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:45_1_1"/><b>then </b>
( <font color="Maroon" title="c25">ff</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="c4">H</font>,<font color="Maroon" title="c3">E</font> &amp; <font color="Maroon" title="c25">ff</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> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E8:45_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="zf_fund1.html#E10:45_1_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="zf_fund1.html#E22:45_1_1"><i><font color="Green" title="E62">A15</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green" title="E22">Lm10</font></i></a></i>;<br/><a NAME="E25:45_1_1"/><b>then </b><i><font color="Green" title="E63">A20</font></i>: 
<font color="Maroon" title="c25">ff</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"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T5" title="ZF_MODEL:th.5">ZF_MODEL:5</a></i>;<br/><font color="Maroon" title="c15">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">ff</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"><span class="p3">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:45_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="zf_fund1.html#E8:45_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="zf_fund1.html#E12:45_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="zf_fund1.html#E21:45_1_1"><i><font color="Green" title="E61">A14</font></i></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">ff</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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T66" title="ZF_LANG1:th.66">ZF_LANG1:66</a></i>
;<br/><b>hence </b><a NAME="E27:45_1_1"/>
<font color="Maroon" title="c15">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"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E25:45_1_1"><i><font color="Green" title="E63">A20</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</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" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E3:45_1"/>
 <a href="zf_fund1.html#K7" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span>,<font color="Maroon" title="c3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="c15">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_1_2"/>
<font color="Maroon" title="c15">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"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span>,<font color="Maroon" title="c3">E</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">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><b> such that </b><br/><a NAME="E3:45_1_2"/><i><font color="Green" title="E53">A21</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">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"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span>,<font color="Maroon" title="c3">E</font> )
 <b>by </b><i><a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></i>;<br/>
<a NAME="E4:45_1_2"/><i><font color="Green" title="E54">A22</font></i>: 
( <font color="Maroon" title="c16">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="c4">H</font>,<font color="Maroon" title="c3">E</font> &amp; <font color="Maroon" title="c16">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> )
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:45_1_2"><i><font color="Green" title="E53">A21</font></i></a>, <a class="ref" href="zf_model.html#T5" title="ZF_MODEL:th.5">ZF_MODEL:5</a></i>;<br/>
<a NAME="E5:45_1_2"/><i><font color="Green" title="E55">A23</font></i>: 
 <a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T66" title="ZF_LANG1:th.66">ZF_LANG1:66</a></i>;<br/>
<b>then </b><i><font color="Green" title="E56">A24</font></i>: <font color="Maroon" title="c15">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:45_1_2"><i><font color="Green" title="E53">A21</font></i></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">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="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="c4">H</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T107" title="RELAT_1:th.107">RELAT_1:107</a></i>
;<br/>
<a NAME="E7:45_1_2"/><i><font color="Green" title="E57">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c4">H</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:45_1_2"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c4">H</font></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="zf_fund1.html#E2:45"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></i>;<br/>
<a NAME="E9:45_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c4">H</font></span>)</span></span>)</span></span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c4">H</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/>
<a NAME="E10:45_1_2"/><b>then </b><i><font color="Green" title="E58">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c4">H</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c4">H</font></span>)</span></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="zf_fund1.html#E4:45"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></i>;<br/>
<a NAME="E12:45_1_2"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c18">y</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:45_1_2"><i><font color="Green" title="E56">A24</font></i></a></i>;<br/>
<a NAME="E13:45_1_2"/><b>then </b><i><font color="Green" title="E59">A27</font></i>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E7:45_1_2"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="zf_fund1.html#E10:45_1_2"><i><font color="Green" title="E58">A26</font></i></a></i>;<br/>
<i><font color="Green" title="E60">A28</font></i>: <font color="Maroon" title="c15">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c4">H</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:45_1_2"><i><font color="Green" title="E53">A21</font></i></a>, <a class="txt" href="zf_fund1.html#E5:45_1_2"><i><font color="Green" title="E55">A23</font></i></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<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>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">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="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></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" title="ZF_FUND1:func.6">Free</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K5" title="ZF_FUND1:func.5">code</a> <span class="p4">(<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>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T107" title="RELAT_1:th.107">RELAT_1:107</a></i>
;<br/>
<a NAME="E15:45_1_2"/><i><font color="Green" title="E61">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:45_1_2"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self" title="ZF_FUND1:def.5">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">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> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="zf_fund1.html#E2:45"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></i>;<br/>
<a NAME="E17:45_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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></span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/>
<a NAME="E18:45_1_2"/><b>then </b><i><font color="Green" title="E62">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">t</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">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"><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="c4">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="zf_fund1.html#E6:45"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></i>;<br/>
<a NAME="E20:45_1_2"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">z</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c20">t</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E14:45_1_2"><i><font color="Green" title="E60">A28</font></i></a></i>;<br/>
<a NAME="E21:45_1_2"/><b>then </b>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E15:45_1_2"><i><font color="Green" title="E61">A29</font></i></a>, <a class="txt" href="zf_fund1.html#E18:45_1_2"><i><font color="Green" title="E62">A30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:45_1_2"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D2</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E13:45_1_2"><i><font color="Green" title="E59">A27</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="ZF_FUND1:func.7">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</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>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:45"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E10:45"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="zf_fund1.html#T5" target="_self" title="ZF_FUND1:th.5">Th5</a></i>;<br/>


</div>
