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<div class="add">

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

<b>thus </b><a NAME="E1:4_1_2"/>
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">x</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">x</font> ) implies <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:4_1_2_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E3:4_1_2_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">x</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:4_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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">x</font> = <font color="Maroon">x</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>
<a NAME="E7:4_1_2_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E8:4_1_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:4_1_2"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/>


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