<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">F</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:4"/><b>then </b>
 <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">F</font>
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then </b><i><font color="Green">E41</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
;<br/>
<a NAME="E4:4"/>
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></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="E1:4_2"/><i><font color="Green">E43</font></i>: 
<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"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E2:4_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E3:4_2"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E4:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E5:4_2"/><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><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:4_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:4"/>
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div>
