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<b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="card_3.html#NM1" title="CARD_3:NM.1">Cardinal-Function</a>;<br/>

<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>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> &amp; $1 is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> );<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:21_3_2"/><i><font color="Green" title="E16">A5</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>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 <b>and </b><br/><a NAME="E2:21_3_2"/><i><font color="Green" title="E17">A6</font></i>: 
for <font color="Olive" title="b1">K</font> being  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>  st <font color="Olive" title="b1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E3:21_3_2"/><i><font color="Green" title="E18">A7</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>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 <b>and </b><br/><a NAME="E4:21_3_2"/><i><font color="Green" title="E19">A8</font></i>: 
for <font color="Olive" title="b1">K</font> being  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>  st <font color="Olive" title="b1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">N</font>
 ;<br/>

<a NAME="E5:21_3_2"/><i><font color="Green" title="E20">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font>
 
<b>from </b><i><a class="ref" href="xboole_0.html#S2" title="XBOOLE_0:sch.2">XBOOLE_0:sch 2</a>(<a class="txt" href="card_5.html#E1:21_3_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="card_5.html#E3:21_3_2"><i><font color="Green" title="E18">A7</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:21_3_2_1"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c5">x</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="c3">f1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">K</font> = <font color="Maroon" title="c5">x</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="card_5.html#E1:21_3_2"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/><a NAME="E3:21_3_2_1"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c6">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_5.html#E1:21_3_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="card_5.html#E1:21_3_2_1"><i><font color="Green" title="E21">A10</font></i></a></i>;<br/><b>hence </b><font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="txt" href="card_5.html#E2:21_3_2"><i><font color="Green" title="E17">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="card_5.html#E4:21_3_2"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="card_5.html#E3:21_3_2_1"><i><font color="Green" title="E22">A11</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:21_3_2"/>
<font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="txt" href="card_5.html#E5:21_3_2"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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