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<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> iff ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">y</font>] ) );<br/>
<a NAME="E1:55_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</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">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><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>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:55_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font>] );<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:55_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">y</font>] )
 <b>and </b><br/><a NAME="E3:55_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y2</font> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">y</font>] )
 ;<br/>

<b>thus </b><a NAME="E4:55_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>from </b><i><a class="ref" href="xboole_0.html#S2">XBOOLE_0:sch 2</a>(, );</i><br/>


</div><b>end;</b></div>
<a NAME="E2:55_1"/><i><font color="Green">E55</font></i>: 
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">F<sub>1</sub></font>() holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_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="E1:55_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font>];<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>) &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E4:55_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">Y</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:55_1"/>
 ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( 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">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );</i><br/>


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