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<b>let </b><font color="Maroon">S</font> be  <a href="fraenkel.html#V1">functional</a> <a href="pralg_2.html#V1">with_common_domain</a>  <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1"/>
( <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>suppose </b></a><a NAME="E1:10_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <font color="Maroon">S</font></span>)</span> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>
<br/>.= 
 <a href="pralg_2.html#K1">DOM</a> <font color="Maroon">S</font>
<b>by </b><i>, <a class="ref" href="pralg_2.html#D2">PRALG_2:def 2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>suppose </b></a><a NAME="E1:10_1_2"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <font color="Maroon">S</font></span>)</span>
 ;<br/><a NAME="E2:10_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:10_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pralg_2.html#K1">DOM</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#D2">PRALG_2:def 2</a></i>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E4:10_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pralg_2.html#K1">DOM</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E5:10_1_2"/><b>then </b>
for <font color="Olive">f</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">S</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font>
 <b>by </b><i><a class="ref" href="pralg_2.html#D2">PRALG_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:10_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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