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

<b>consider </b><font color="Maroon" title="c1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(),   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()] <b>means </b>( (  ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b1">c</font>] implies <font color="Maroon">P<sub>1</sub></font>[$1,$2] ) &amp; ( ( for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  holds <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b1">c</font>] ) implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) );<br/>
<a NAME="E2:18_1"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  ex <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:18_1_1"/>
( ( for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  holds <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Olive" title="b1">c</font>] ) implies  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( (  ex <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Olive" title="b2">c</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Olive" title="b1">z</font>] ) &amp; ( ( for <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  holds <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Olive" title="b2">c</font>] ) implies <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) ) )
 
;<br/>
<b>hence </b><a NAME="E2:18_1_1"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">d</font>,<font color="Olive" title="b1">z</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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="E4:18_1"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">d</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="seq_1.html#E2:18_1"><i><font color="Green" title="E7">A1</font></i></a>);<br/></i>
<a NAME="E5:18_1"/><i><font color="Green" title="E9">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b1">c</font>];<br/>
<b>consider </b><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:18_1"/><i><font color="Green" title="E10">A4</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> <font color="Maroon" title="c3">X</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; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E8:18_1"/>
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> <font color="Maroon" title="c3">X</font> holds <br/><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>()
 
<b>by </b><i><a class="txt" href="seq_1.html#E7:18_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c3">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font> as   <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>() ;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<b>thus </b><a NAME="E11:18_1"/>
for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive" title="b1">d</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="c5">f</font> iff  ex <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">c</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:18_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <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="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/>
<b>hence </b><a NAME="E2:18_1_2"/>
( <font color="Maroon" title="c6">d</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="c5">f</font> implies  ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">d</font>,<font color="Olive" title="b1">c</font>] )
 <b>by </b><i><a class="txt" href="seq_1.html#E7:18_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>

<b>assume </b><a NAME="E3:18_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">d</font>,<font color="Olive" title="b1">c</font>]
 ;<br/>

<a NAME="E4:18_1_2"/><b>then </b>
( <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="seq_1.html#E7:18_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>
<a NAME="E5:18_1_2"/><b>then </b>
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="seq_1.html#E5:18_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_2"/>
<font color="Maroon" title="c6">d</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="c5">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c6">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E12:18_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c6">d</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="c5">f</font>
 ;<br/>

<a NAME="E13:18_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <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="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/>
<a NAME="E14:18_1"/><b>then </b>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">d</font>,<font color="Olive" title="b1">c</font>]
 
<b>by </b><i><a class="txt" href="seq_1.html#E7:18_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="seq_1.html#E12:18_1"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<a NAME="E15:18_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font>]
 
<b>by </b><i><a class="txt" href="seq_1.html#E4:18_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:18_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">d</font>]
 <b>by </b><i><a class="txt" href="seq_1.html#E12:18_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div>
