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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">a'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b2">t</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">F</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Olive" title="b1">a'</font>,<font color="Maroon" title="c3">G</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Olive" title="b1">a'</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a'</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> );<br/>
<div><i><font color="Green" title="E8">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><b>consider </b><font color="Maroon" title="c6">f</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">F</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c3">G</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c5">a'</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">f'</font> = <font color="Maroon" title="c6">f</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c8">f'</font> = <font color="Maroon" title="c7">f'</font>;<br/><b>thus </b><a NAME="E5:28_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">f'</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:28_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="index_1.html#E1:28_1_1"><i><font color="Green" title="E8">A1</font></i></a>);<br/></i>
<a NAME="E4:28_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:28_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<a NAME="E2:28_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">a'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b2">t</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">F</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Olive" title="b1">a'</font>,<font color="Maroon" title="c3">G</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Olive" title="b1">a'</font> st <br/>( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a'</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> )
 
<b>by </b><i><a class="txt" href="index_1.html#E3:28_1_1"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:28_1_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="index_1.html#E3:28_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E6:28_1_1"/>
 ex <font color="Olive" title="b1">a'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b2">t</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">F</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Olive" title="b1">a'</font>,<font color="Maroon" title="c3">G</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Olive" title="b1">a'</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a'</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> )
 
<b>by </b><i><a class="txt" href="index_1.html#E3:28_1_1"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:28_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> is    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">F</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">G</font> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c6">a</font>
 ;<br/>


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