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

<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>



<a NAME="E1:48"/>
for <font color="Olive">F</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">F</font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">Y</font> holds <br/>for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> holds <br/><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:48_1"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> :  ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="partit1.html#T35">PARTIT1:35</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:48_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:48_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E7:48_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E8:48_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:48_1"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:48"/>
<font color="Maroon">a</font> <a href="bvfunc_1.html#R2" target="_self">is_dependent_of</a>  <a href="partit1.html#NK6" target="_self">%I</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div>
