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<div class="add">

<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/><b>let </b><font color="Maroon">f</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="E1:12"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> ;<br/>
<i><font color="Green">E45</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> = 
 <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/>.= 
 <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/>
;<br/>
<a NAME="E4:12"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</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> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</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="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon">i</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="E2:12_2"/><b>then </b>
<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="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:12_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</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="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:12"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</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>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div>
