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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>) );<br/>
<a NAME="E1:20_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="partfun1.html#E1:20"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E2:20_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) )
 <b>and </b><br/><a NAME="E5:20_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="partfun1.html#S2" target="_self">PARTFUN1:sch 2</a>(<a class="txt" href="partfun1.html#E1:20_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="partfun1.html#E2:20_1"><i><font color="Green">E13</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>thus </b><a NAME="E6:20_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:20_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> implies ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:20_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) ) )
 
<b>by </b><i><a class="txt" href="partfun1.html#E4:20_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 ;<br/>


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

<b>assume </b><a NAME="E2:20_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 ;<br/>

<a NAME="E3:20_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) ) )
 
;<br/>
<b>hence </b><a NAME="E4:20_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="partfun1.html#E4:20_1"><i><font color="Green">E14</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E7:20_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>by </b><i><a class="txt" href="partfun1.html#E5:20_1"><i><font color="Green">E15</font></i></a></i>;<br/>


</div>
