<?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">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">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="c4">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> ;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> $1,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>;<br/>
<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c3">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> <font color="Maroon" title="c3">X</font> )
 ;<br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c3">X</font> is <a href="zf_fund1.html#V5" title="ZF_FUND1:attr.5">closed_wrt_A5</a>
 
<b>by </b><i><a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></i>;<br/>
<a NAME="E3:29"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c3">X</font> is <a href="zf_fund1.html#V4" title="ZF_FUND1:attr.4">closed_wrt_A4</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><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/>
<a NAME="E4:29"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c4">fs</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<a NAME="E5:29"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_1"/><i><font color="Green" title="E39">A6</font></i>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T64" title="FUNCT_5:th.64">FUNCT_5:64</a></i>;<br/>
<a NAME="E2:29_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T3" target="_self" title="ZF_FUND1:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E1:29_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:29"/><i><font color="Green" title="E39">A7</font></i>: 
for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">fs</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">fs</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green" title="E40">A8</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">fs</font> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">fs</font> &amp;  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_1"/>
( <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1"/>
<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></i>;<br/><b>hence </b><a NAME="E3:29_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29_2"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_2"/><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_2_1_2"/><i><font color="Green" title="E42">A10</font></i>: 
 <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="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T7" target="_self" title="ZF_FUND1:th.7">Th7</a></i>;<br/><a NAME="E3:29_2_1_2"/><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29_2"><i><font color="Green" title="E40">A8</font></i></a></i>;<br/><a NAME="E4:29_2_1_2"/><b>then </b><i><font color="Green" title="E44">A12</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:29_2_1_2"><i><font color="Green" title="E42">A10</font></i></a></i>;<br/><a NAME="E5:29_2_1_2"/><i><font color="Green" title="E45">A13</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E2:29_2_1_2"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="zf_fund1.html#E3:29_2_1_2"><i><font color="Green" title="E43">A11</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="c7">r</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c5">o</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#E4:29_2_1_2"><i><font color="Green" title="E44">A12</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c9">A</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> ;<br/><a NAME="E7:29_2_1_2"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_2_1_2_1"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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><br/>

<b>assume </b><a NAME="E1:29_2_1_2_1_1"/>
<font color="Maroon" title="c10">q</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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:29_2_1_2_1_1"/><i><font color="Green" title="E46">A14</font></i>: 
( <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</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/>
<a NAME="E4:29_2_1_2_1_1"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">g</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_1_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:29_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E47">A15</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">g</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:29_2_1_2_1_1"/><i><font color="Green" title="E48">A16</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI: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>reconsider </b><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</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#E1:29"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E3:29_2_1_2_1_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="zf_fund1.html#E5:29_2_1_2_1_1"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></i>;<br/>
<a NAME="E8:29_2_1_2_1_1"/>
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c12">s</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_1_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="grfunc_1.html#T18" title="GRFUNC_1:th.18">GRFUNC_1:18</a></i>;<br/>
<b>hence </b><a NAME="E9:29_2_1_2_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_1_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="zf_fund1.html#E5:29_2_1_2_1_1"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="zf_fund1.html#E6:29_2_1_2_1_1"><i><font color="Green" title="E48">A16</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c10">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><br/>

<b>assume </b><a NAME="E2:29_2_1_2_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </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="E4:29_2_1_2_1"/><i><font color="Green" title="E46">A17</font></i>: 
( <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c10">q</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#E4:29_2_1_2_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="grfunc_1.html#T15" title="GRFUNC_1:th.15">GRFUNC_1:15</a></i>;<br/>
<a NAME="E6:29_2_1_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:29_2_1_2_1"/><b>then </b><i><font color="Green" title="E47">A18</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/>
<a NAME="E8:29_2_1_2_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E9:29_2_1_2_1"/>
<font color="Maroon" title="c10">q</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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E7:29_2_1_2_1"><i><font color="Green" title="E47">A18</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:29_2_1_2"/><b>then </b><i><font color="Green" title="E46">A19</font></i>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E3:29"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="zf_fund1.html#E5:29_2_1_2"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="zf_fund1.html#D9" target="_self" title="ZF_FUND1:def.9">Def9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">x</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c2">a</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#E1:29_2"><i><font color="Green" title="E40">A8</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">y</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</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#E8:29_2_1_2"><i><font color="Green" title="E46">A19</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">Z</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="c10">x</font> &amp; <font color="Olive" title="b2">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span>}</span> ;<br/><a NAME="E11:29_2_1_2"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font> <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">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="c10">x</font> &amp; <font color="Olive" title="b2">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_2_1_2_2"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</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="c10">x</font> &amp; <font color="Olive" title="b2">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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><br/>

<b>assume </b><a NAME="E1:29_2_1_2_2_1"/>
<font color="Maroon" title="c13">q</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"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:29_2_1_2_2_1"/><i><font color="Green" title="E47">A20</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</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>set </b><font color="Maroon" title="c15">A</font> = <font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font>;<br/>
<b>set </b><font color="Maroon" title="c16">C</font> = <font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<i><font color="Green" title="E48">A21</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_2_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
;<br/>
<a NAME="E5:29_2_1_2_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<a NAME="E6:29_2_1_2_2_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_2_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:29_2_1_2_2_1"/><b>then </b><i><font color="Green" title="E49">A22</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_2_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">x'</font> = <font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">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#E1:29_2"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></i>;<br/>
<i><font color="Green" title="E50">A23</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_2_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
;<br/>
<a NAME="E10:29_2_1_2_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<a NAME="E11:29_2_1_2_2_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_2_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:29_2_1_2_2_1"/><b>then </b><i><font color="Green" title="E51">A24</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E9:29_2_1_2_2_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">y'</font> = <font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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#E8:29_2_1_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></i>;<br/>
<font color="Maroon" title="c14">g</font> = 
<font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_2_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="E15:29_2_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c13">q</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#E3:29_2_1_2_2_1"><i><font color="Green" title="E47">A20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:29_2_1_2_2_1"/>
<font color="Maroon" title="c13">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">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="c10">x</font> &amp; <font color="Olive" title="b2">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E7:29_2_1_2_2_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="zf_fund1.html#E12:29_2_1_2_2_1"><i><font color="Green" title="E51">A24</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c13">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><br/>

<b>assume </b><a NAME="E2:29_2_1_2_2"/>
<font color="Maroon" title="c13">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">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="c10">x</font> &amp; <font color="Olive" title="b2">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">x'</font>, <font color="Maroon" title="c15">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="E4:29_2_1_2_2"/><i><font color="Green" title="E47">A25</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c15">y'</font> &amp; <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c15">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c16">e</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:29_2_1_2_2"/><i><font color="Green" title="E48">A26</font></i>: 
( <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">e</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_2"><i><font color="Green" title="E47">A25</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="c17">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:29_2_1_2_2"/><i><font color="Green" title="E49">A27</font></i>: 
( <font color="Maroon" title="c15">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_2"><i><font color="Green" title="E47">A25</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c16">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">g</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#E1:29_2_1_2"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="zf_fund1.html#E6:29_2_1_2_2"><i><font color="Green" title="E48">A26</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2_2"><i><font color="Green" title="E49">A27</font></i></a>, <a class="ref" href="grfunc_1.html#T31" title="GRFUNC_1:th.31">GRFUNC_1:31</a></i>;<br/>
<a NAME="E10:29_2_1_2_2"/><i><font color="Green" title="E50">A28</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:29_2_1_2_2"><i><font color="Green" title="E48">A26</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2_2"><i><font color="Green" title="E49">A27</font></i></a>, <a class="ref" href="grfunc_1.html#T33" title="GRFUNC_1:th.33">GRFUNC_1:33</a></i>;<br/>
<a NAME="E11:29_2_1_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">e</font></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="c17">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T33" title="GRFUNC_1:th.33">GRFUNC_1:33</a></i>;<br/>
<a NAME="E12:29_2_1_2_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:29_2_1_2_2"><i><font color="Green" title="E48">A26</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2_2"><i><font color="Green" title="E49">A27</font></i></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/>
<b>hence </b><a NAME="E13:29_2_1_2_2"/>
<font color="Maroon" title="c13">q</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"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_2"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="zf_fund1.html#E6:29_2_1_2_2"><i><font color="Green" title="E48">A26</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2_2"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="zf_fund1.html#E10:29_2_1_2_2"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:29_2_1_2"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:29"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="zf_fund1.html#E1:29_2"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="zf_fund1.html#D10" target="_self" title="ZF_FUND1:def.10">Def10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E7:29"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">fs</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="zf_fund1.html#E4:29"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="zf_fund1.html#E5:29"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="zf_fund1.html#E6:29"><i><font color="Green" title="E39">A7</font></i></a>);</i><br/>


</div>
