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

<b>thus </b><a NAME="E1:3_1_2"/>
( not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> iff 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="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> ) ) &amp; ( 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> <font color="Maroon">A</font> holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<font color="Olive">i</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> iff 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="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> ) ) &amp; ( 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> <font color="Maroon">B</font> holds <br/><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<font color="Olive">i</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="3_1_2_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </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">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:3_1_2_1"/>
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 <b>and </b><br/><a NAME="E2:3_1_2_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] )
 <b>and </b><br/><a NAME="E3:3_1_2_1"/><i><font color="Green">E63</font></i>: 
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> <font color="Maroon">A</font> holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<font color="Olive">i</font>
 <b>and </b><br/><a NAME="E4:3_1_2_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] )
 <b>and </b><br/><a NAME="E5:3_1_2_1"/><i><font color="Green">E65</font></i>: 
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> <font color="Maroon">B</font> holds <br/><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<font color="Olive">i</font>
 ;<br/>

<a NAME="E6:3_1_2_1"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
 
<b>from </b><i><a class="ref" href="xboole_0.html#S2">XBOOLE_0:sch 2</a>(, );<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1_1"><b>now </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="E1:3_1_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font>
 ;<br/><b>hence </b><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="card_3.html#K5">pi</a> <font color="Maroon">S</font>,<font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:3_1_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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

<b>thus </b><a NAME="E2:3_1_2"/>
( <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</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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