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

<b>thus </b><a NAME="E1:4_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1_1_1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="pralg_2.html#E3:4_1_1_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>
<b>take </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E5:4_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E6:4_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:4_1_1_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="pralg_2.html#D1" target="_self">Def1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:4_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</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>
