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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> ] <b>means </b> ex <font color="Olive" title="b1">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">g</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b1">g</font> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c4">F</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:101_1_1_1"/><i><font color="Green" title="E47">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i><b>take </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">F</font>;<br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E3:101_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> implies  ex <font color="Olive" title="b1">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive" title="b1">g</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b1">g</font> ) )
 <b>by </b><i><a class="txt" href="partfun1.html#E2:101_1_1_1"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/><b>given </b><font color="Maroon" title="c7">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><b> such that </b><a NAME="E5:101_1_1_1"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>and </b><br/><a NAME="E6:101_1_1_1"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c7">g</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>and </b><br/><a NAME="E7:101_1_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c7">g</font>
 ;<br/><a NAME="E8:101_1_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="partfun1.html#T119" target="_self" title="PARTFUN1:th.119">Th119</a></i>;<br/><b>hence </b><a NAME="E9:101_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="partfun1.html#E2:101_1_1_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="partfun1.html#E5:101_1_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="partfun1.html#E6:101_1_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="partfun1.html#E7:101_1_1_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:101_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> iff  ex <font color="Olive" title="b3">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b3">g</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b3">g</font> ) )
 ;<br/>


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