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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="membered.html#V2" title="MEMBERED:attr.2">real-membered</a> <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="membered.html#T2" title="MEMBERED:th.2">MEMBERED:2</a></i>;<br/>
<a NAME="E3:6"/><i><font color="Green" title="E5">A2</font></i>: 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:6"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="pscomp_1.html#T32" title="PSCOMP_1:th.32">PSCOMP_1:32</a></i>;<br/>
<a NAME="E4:6"/><i><font color="Green" title="E6">A3</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal6.html#T78" title="TOPREAL6:th.78">TOPREAL6:78</a></i>;<br/>
<a NAME="E5:6"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T30" title="RCOMP_1:th.30">RCOMP_1:30</a></i>;<br/>
<b>hence </b><a NAME="E6:6"/>
 <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:6"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="rcomp_3.html#E4:6"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>


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