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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">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Olive" title="b1">x</font> );<br/>
<b>consider </b><font color="Maroon" title="c2">Z</font> being   <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> <b> such that </b><br/><a NAME="E2:30_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">z</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] )
 <b>from </b><i><a class="ref" href="supinf_1.html#S1" title="SUPINF_1:sch.1">SUPINF_1:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c3">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E4:30_1_1"/>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c3">x</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">Z</font> = <font color="Maroon" title="c2">Z</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#K7" title="NUMBERS:func.7">ExtREAL</a>  <b>by </b><i><a class="txt" href="supinf_2.html#E2:30_1_1"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">Z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> iff  ex <font color="Olive" title="b2">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Olive" title="b2">x</font> ) )</span><br/>

<b>thus </b><a NAME="E6:30_1_1"/>
for <font color="Olive" title="b1">z</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> iff  ex <font color="Olive" title="b2">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Olive" title="b2">x</font> ) )
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:30_1_1"><i><font color="Green" title="E26">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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