<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">X</font> being  <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b4">fs</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b3">X</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> &amp; <font color="Olive" title="b1">a</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> <font color="Olive" title="b4">fs</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">X</font> holds <br/><span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b5">x</font></span>)</span> where <font color="Olive" title="b5">x</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="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">b</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">X</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">fs</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">X</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> &amp; <font color="Maroon" title="c2">a</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> <font color="Olive" title="b2">fs</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/><span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b3">x</font></span>)</span> where <font color="Olive" title="b3">x</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="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font></span><br/><b>let </b><font color="Maroon" title="c4">X</font> be   <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">fs</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Maroon" title="c4">X</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> &amp; <font color="Maroon" title="c2">a</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> <font color="Olive" title="b1">fs</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b2">x</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="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><b>let </b><font color="Maroon" title="c5">fs</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> &amp; <font color="Maroon" title="c2">a</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> <font color="Maroon" title="c5">fs</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/>









<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c4">X</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> &amp; <font color="Maroon" title="c2">a</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> <font color="Maroon" title="c5">fs</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/>

<a NAME="E2:30"/><b>then </b>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c5">fs</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="ref" href="zf_fund1.html#T8" target="_self" title="ZF_FUND1:th.8">Th8</a></i>;<br/>
<a NAME="E3:30"/><b>then </b><i><font color="Green" title="E35">A2</font></i>: 
<span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c2">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="c4">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:30"><i><font color="Green" title="E34">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="c6">s</font> = <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c8">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span> ;<br/>
<a NAME="E4:30"/><i><font color="Green" title="E36">A3</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span>  <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">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span> 
  <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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span>  <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="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">q</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">q</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="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span>  or <font color="Maroon" title="c9">q</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">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c9">q</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="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">q</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">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span> </span><br/>

<a NAME="E2:30_1_1"/><b>then </b><i><font color="Green" title="E37">A4</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> )
 
;<br/>
<a NAME="E3:30_1_1"/>
<font color="Maroon" title="c2">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="c2">a</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="E4:30_1_1"/>
<font color="Maroon" title="c9">q</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">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:30_1_1"><i><font color="Green" title="E37">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:30_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span>  <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="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">q</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">q</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">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span>  or <font color="Maroon" title="c9">q</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="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:30_1_2"/>
<font color="Maroon" title="c9">q</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">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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">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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">q</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="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c11">x</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="E3:30_1_2"/><i><font color="Green" title="E37">A5</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c11">x</font> &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="c2">a</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> )
 ;<br/>
<a NAME="E4:30_1_2"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:30_1_2"><i><font color="Green" title="E37">A5</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="E5:30_1_2"/>
<font color="Maroon" title="c9">q</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="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:30_1_2"><i><font color="Green" title="E37">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<a NAME="E5:30"/>
<font color="Maroon" title="c4">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#E1:30"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E6:30"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="zf_fund1.html#K2" title="ZF_FUND1:func.2">(#)</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</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="c3">b</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:30"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E3:30"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E4:30"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="zf_fund1.html#D12" target="_self" title="ZF_FUND1:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
