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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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green" title="E18">A2</font></i>: 
not <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:18"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c2">r</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="seq_4.html#E2:18"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )
 
<b>by </b><i><a class="txt" href="seq_4.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="seq_4.html#D3" target="_self" title="SEQ_4:def.3">Def3</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green" title="E21">A5</font></i>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E4:18"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="seq_4.html#D5" target="_self" title="SEQ_4:def.5">Def5</a></i>;<br/>
<a NAME="E7:18"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E4:18"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="seq_4.html#E5:18"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="seq_4.html#D4" target="_self" title="SEQ_4:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:18"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E6:18"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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