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<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:101"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Y</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>and </b><br/><a NAME="E2:101"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:101"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:101_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:101_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> )
 <b>by </b><i><a class="ref" href="complsp1.html#T51" target="_self">Th51</a></i>;<br/><a NAME="E4:101_1"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T49" target="_self">Th49</a>, <a class="ref" href="complsp1.html#T52" target="_self">Th52</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>hence </b><a NAME="E5:101_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T52" target="_self">Th52</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:101"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T50" target="_self">Th50</a>, <a class="txt" href="complsp1.html#E49"><i><font color="Green">Lemma67</font></i></a></i>;<br/>


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