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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#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> implies (  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<a NAME="E3:21"/>
( <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> )
 
<b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T28" target="_self" title="RCOMP_1:th.28">Th28</a></i>;<br/>
<a NAME="E4:21"/><b>then </b>
( <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> )
 
;<br/>
<b>hence </b><a NAME="E5:21"/>
(  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="rcomp_1.html#T30" target="_self" title="RCOMP_1:th.30">Th30</a>, <a class="ref" href="rcomp_1.html#T31" target="_self" title="RCOMP_1:th.31">Th31</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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