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<div class="add">

<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a>;<br/>
<b>take </b><font color="Maroon">A</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">h</font></span><a href="tarski.html#K1">}</a></span>;<br/>

<a NAME="E2:2_1_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</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">A</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:2_1_1_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:2_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:2_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_1_1"/>
<font color="Maroon">A</font> is <a href="pralg_2.html#V1" target="_self">with_common_domain</a>
 <b>by </b><i/>;<br/>

<a NAME="E4:2_1_1"/>
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> <font color="Maroon">A</font> holds <br/><font color="Olive">x</font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:2_1_1"/>
<font color="Maroon">A</font> is <a href="fraenkel.html#V1">functional</a>
 <b>by </b><i><a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>

<b>thus </b><a NAME="E6:2_1_1"/>
not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


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