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<span class="kw">consider </span><font color="Maroon" title="c2">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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">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/>
<span class="kw">consider </span><font color="Maroon" title="c3">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> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
for <font color="Olive" title="b1">z</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">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="c3">Z</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></span>
<a NAME="E4:26_1_1"/>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c2">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="c2">x</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">Z</font> = <font color="Maroon" title="c3">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">Z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">z</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">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#NM1" title="SUPINF_1:NM.1">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/>

<span class="kw">thus </span><a NAME="E6:26_1_1"/>
for <font color="Olive" title="b1">z</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">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#NM1" title="SUPINF_1:NM.1">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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