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





<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E16</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 ;<br/>

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>

<a NAME="E3:33"/><i><font color="Green">E19</font></i>: 
 ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">phi</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E6:33"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f'</font> holds <br/><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T136">FUNCT_2:136</a></i>;<br/>
<b>reconsider </b><font color="Maroon">phi</font> = <font color="Maroon">phi</font> as    <a href="fraenkel.html#M2" target="_self">Element</a> of  <a href="fraenkel.html#K1" target="_self">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">phi</font>
;<br/>

 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f'</font> = 
<font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">phi</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E9:33"/>
<font color="Maroon">phi</font> <a href="relset_1.html#K8">|</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


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