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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</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/>
<a NAME="E1:87_1_1"/><i><font color="Green" title="E57">A1</font></i>: 
( <font color="Maroon" title="c1">F</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  &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">F</font> holds <br/><font color="Olive" title="b1">A</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ) )
 
<b>by </b><i><a class="ref" href="supinf_1.html#D18" target="_self" title="SUPINF_1:def.18">Def18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">F</font>;<br/>
<b>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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  <b>by </b><i><a class="txt" href="supinf_1.html#E1:87_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:87_1_1"/><i><font color="Green" title="E58">A2</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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E6:87_1_1"/><i><font color="Green" title="E59">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</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="supinf_1.html#S1" target="_self" title="SUPINF_1:sch.1">SUPINF_1:sch 1</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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="supinf_1.html#E1:87_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="supinf_1.html#E4:87_1_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="supinf_1.html#E6:87_1_1"><i><font color="Green" title="E59">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">S</font> = <font color="Maroon" title="c5">S</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c6">S</font>
;<br/>

<b>thus </b><a NAME="E9:87_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</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="c6">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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</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#E6:87_1_1"><i><font color="Green" title="E59">A3</font></i></a></i>;<br/>


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