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

<b>thus </b><a NAME="E1:3_1_1"/>
( not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a> implies  ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> iff for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> ) ) &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> holds <br/><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<font color="Olive">i</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1_1"/>
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">S</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font> : verum </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:3_1_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font> : verum </span>}</span>  holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>]
 
;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:3_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E6:3_1_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font> : verum </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><a NAME="E2:3_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E3:3_1_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font> : verum </span>}</span> 
 ;<br/><b>hence </b><a NAME="E4:3_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:3_1_1_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font>
 ;<br/><b>consider </b><font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font>;<br/><a NAME="E4:3_1_1_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font> : verum </span>}</span> 
 ;<br/><a NAME="E5:3_1_1_1_1"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font> : verum </span>}</span>  holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_1_1_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:3_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S1</font> st <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font>
 
;<br/>
<b>hence </b><a NAME="E3:3_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E8:3_1_1_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<font color="Olive">i</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E2:3_1_1"/>
( <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a> implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>


</div>
