<?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">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E28">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">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">A</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="c6">b</font> = <font color="Maroon" title="c4">B</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#E1:24"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E4:24"/><i><font color="Green" title="E29">A2</font></i>: 
( <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></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#E1:24"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></i>;<br/>
<a NAME="E5:24"/><i><font color="Green" title="E30">A3</font></i>: 
<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#E1:24"><i><font color="Green" title="E28">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="E6:24"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="zf_fund1.html#V6" title="ZF_FUND1:attr.6">closed_wrt_A6</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green" title="E28">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="E7:24"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="zf_fund1.html#V7" title="ZF_FUND1:attr.7">closed_wrt_A7</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">D</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span> ;<br/>
<a NAME="E8:24"/><i><font color="Green" title="E33">A6</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </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#E4:24"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E5:24"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="zf_fund1.html#D10" target="_self" title="ZF_FUND1:def.10">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><i><font color="Green" title="E34">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c8">o</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">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="E3:24_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
( <font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> )
 ;<br/><a NAME="E4:24_1_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1"/>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_1_1"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E2:24_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:24_1_2"/>
<font color="Maroon" title="c8">o</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_1_2"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:24_1"/>
( <font color="Maroon" title="c8">o</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span>  iff <font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_1"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:24"/><b>then </b>
<span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></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#E8:24"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:24"/>
<font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">B</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#E1:24"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></i>;<br/>

<b>set </b><font color="Maroon" title="c8">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="classes2.html#K9" title="CLASSES2:func.9">\</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span> ;<br/>
<a NAME="E12:24"/><i><font color="Green" title="E34">A10</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#K9" title="CLASSES2:func.9">\</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </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#E4:24"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E6:24"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="zf_fund1.html#D11" target="_self" title="ZF_FUND1:def.11">Def11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><i><font color="Green" title="E35">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_2_1"/>
<font color="Maroon" title="c9">o</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#K9" title="CLASSES2:func.9">\</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">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="E3:24_2_1"/><i><font color="Green" title="E36">A12</font></i>: 
( <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="classes2.html#K9" title="CLASSES2:func.9">\</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> )
 ;<br/><a NAME="E4:24_2_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_2_1"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:24_2_1"/>
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="classes2.html#K9" title="CLASSES2:func.9">\</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_2_1"><i><font color="Green" title="E36">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_2_2"/><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="classes2.html#K9" title="CLASSES2:func.9">\</a> <font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E2:24_2_2"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:24_2_2"/>
<font color="Maroon" title="c9">o</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#K9" title="CLASSES2:func.9">\</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_2_2"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:24_2"/>
( <font color="Maroon" title="c9">o</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#K9" title="CLASSES2:func.9">\</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span>  iff <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="classes2.html#K9" title="CLASSES2:func.9">\</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_2"><i><font color="Green" title="E35">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:24"/><b>then </b>
<span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="classes2.html#K9" title="CLASSES2:func.9">\</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></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#E12:24"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:24"/>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">B</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#E1:24"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></i>;<br/>

<b>set </b><font color="Maroon" title="c9">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span> ;<br/>
<a NAME="E16:24"/><i><font color="Green" title="E35">A14</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </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#E4:24"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E7:24"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="zf_fund1.html#D12" target="_self" title="ZF_FUND1:def.12">Def12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><i><font color="Green" title="E36">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_3_1"/>
<font color="Maroon" title="c10">o</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="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">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="E3:24_3_1"/><i><font color="Green" title="E37">A16</font></i>: 
( <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> )
 ;<br/><a NAME="E4:24_3_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_3_1"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:24_3_1"/>
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_3_1"><i><font color="Green" title="E37">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_3_2"/><i><font color="Green" title="E37">A17</font></i>: 
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E2:24_3_2"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:24_3_2"/>
<font color="Maroon" title="c10">o</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="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_3_2"><i><font color="Green" title="E37">A17</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:24_3"/>
( <font color="Maroon" title="c10">o</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="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</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> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> ) </span>}</span>  iff <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_3"><i><font color="Green" title="E36">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:24"/><b>then </b>
<span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></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#E16:24"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E19:24"/>
<font color="Maroon" title="c3">A</font> <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> <font color="Maroon" title="c4">B</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#E1:24"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></i>;<br/>


</div>
