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<b>let </b><font color="Maroon" title="c1">X</font> be   <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>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>$1 is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font>;<br/>
<b>consider </b><font color="Maroon" title="c2">Y</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="E2:34"/><i><font color="Green" title="E22">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</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="c2">Y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</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>
<a NAME="E3:34"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T36" target="_self" title="SUPINF_1:th.36">Th36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">Y</font> = <font color="Maroon" title="c2">Y</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#E2:34"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">Y</font> = <font color="Maroon" title="c3">Y</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="c4">Y</font>
;<br/>

<b>thus </b><a NAME="E6:34"/>
for <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</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">Y</font> iff <font color="Olive" title="b1">x</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E2:34"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>


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