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

<b>set </b><font color="Maroon" title="c1">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> 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">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">w</font>,<font color="Olive" title="b1">x</font>] &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() ) </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1,$2] &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() );<br/>
<a NAME="E1:46_1"/><i><font color="Green" title="E13">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 
;<br/>
<a NAME="E2:46_1"/><i><font color="Green" title="E14">A4</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E2:46"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E4:46_1"/><i><font color="Green" title="E15">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>and </b><br/><a NAME="E5:46_1"/><i><font color="Green" title="E16">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="partfun1.html#S2" title="PARTFUN1:sch.2">PARTFUN1:sch 2</a>(<a class="txt" href="fraenkel.html#E1:46_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="fraenkel.html#E2:46_1"><i><font color="Green" title="E14">A4</font></i></a>);<br/></i>
<a NAME="E6:46_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> 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">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">w</font>,<font color="Olive" title="b1">x</font>] &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> 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">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">w</font>,<font color="Olive" title="b1">x</font>] &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>()
  <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="46_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</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="E1:46_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> 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">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">w</font>,<font color="Olive" title="b1">x</font>] &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:46_1_1_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font>
 <b>and </b><br/><a NAME="E4:46_1_1_1"/><i><font color="Green" title="E18">A8</font></i>: 
 ex <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">w</font>,<font color="Maroon" title="c4">u</font>] &amp; <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() )
 ;<br/>
<b>consider </b><font color="Maroon" title="c5">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:46_1_1_1"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c4">u</font>]
 <b>and </b><br/><a NAME="E7:46_1_1_1"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="txt" href="fraenkel.html#E4:46_1_1_1"><i><font color="Green" title="E18">A8</font></i></a></i>;<br/>
<a NAME="E8:46_1_1_1"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E4:46_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="fraenkel.html#E6:46_1_1_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="fraenkel.html#E7:46_1_1_1"><i><font color="Green" title="E20">A10</font></i></a></i>;<br/>
<a NAME="E9:46_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">w</font>]
 
<b>by </b><i><a class="txt" href="fraenkel.html#E5:46_1"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/>
<a NAME="E10:46_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="fraenkel.html#E2:46"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="fraenkel.html#E3:46_1_1_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="fraenkel.html#E6:46_1_1_1"><i><font color="Green" title="E19">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:46_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="txt" href="fraenkel.html#E7:46_1_1_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="fraenkel.html#E8:46_1_1_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c3">x</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="E2:46_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon">F<sub>3</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:46_1_1"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E5:46_1_1"/><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>and </b><br/><a NAME="E6:46_1_1"/><i><font color="Green" title="E19">A14</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="fraenkel.html#E5:46_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="fraenkel.html#E4:46_1_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="fraenkel.html#E6:46_1_1"><i><font color="Green" title="E19">A14</font></i></a></i>;<br/>
<a NAME="E8:46_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">x</font>]
 
<b>by </b><i><a class="txt" href="fraenkel.html#E5:46_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="fraenkel.html#E4:46_1_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="fraenkel.html#E6:46_1_1"><i><font color="Green" title="E19">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:46_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> 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">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">w</font>,<font color="Olive" title="b1">x</font>] &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() ) </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E4:46_1_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="fraenkel.html#E5:46_1_1"><i><font color="Green" title="E18">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:46_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> 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">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">w</font>,<font color="Olive" title="b1">x</font>] &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="fraenkel.html#E1:46"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="finset_1.html#T17" title="FINSET_1:th.17">FINSET_1:17</a></i>;<br/>


</div>
