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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b> ex <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Olive" title="b1">A</font> );<br/>
<b>consider </b><font color="Maroon" title="c2">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">F</font>;<br/>
<a NAME="E2:84_1_1"/>
<font color="Maroon" title="c2">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="setfam_1.html#D9" title="SETFAM_1:def.9">SETFAM_1:def 9</a></i>;<br/>
<a NAME="E4:84_1_1"/><i><font color="Green" title="E51">A1</font></i>: 
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c3">A</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c4">S</font> being  <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:84_1_1"/><i><font color="Green" title="E52">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )
 <b>from </b><i><a class="ref" href="membered.html#S2" title="MEMBERED:sch.2">MEMBERED:sch 2</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">S</font> = <font color="Maroon" title="c4">S</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="supinf_1.html#E4:84_1_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="supinf_1.html#E6:84_1_1"><i><font color="Green" title="E52">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">S</font>
;<br/>

<b>thus </b><a NAME="E8:84_1_1"/>
for <font color="Olive" title="b1">a</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font> iff  ex <font color="Olive" title="b2">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Olive" title="b2">A</font> ) )
 <b>by </b><i><a class="txt" href="supinf_1.html#E6:84_1_1"><i><font color="Green" title="E52">A2</font></i></a></i>;<br/>


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