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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="supinf_1.html#NM3">R_eal</a>] <b>means </b> ex <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Olive">A</font> );<br/>
<a NAME="E1:87_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">F</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K6">ExtREAL</a>  &amp; ( for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">A</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ) )
 
<b>by </b><i><a class="ref" href="supinf_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F</font>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:87_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">A</font>
 
;<br/>
<b>consider </b><font color="Maroon">S</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E6:87_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a> holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] )
 <b>from </b><i><a class="ref" href="supinf_1.html#S1" target="_self">SUPINF_1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="supinf_1.html#T55" target="_self">Th55</a>, <a class="ref" href="supinf_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  ;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E9:87_1_1"/>
for <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a> holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff  ex <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Olive">A</font> ) )
 <b>by </b><i><a class="ref" href="supinf_1.html#D15" target="_self">Def15</a></i>;<br/>


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