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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> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:86"/><i><font color="Green">E59</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="E2:86"/><i><font color="Green">E60</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/><font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r'</font>
 ;<br/>

<a NAME="E3:86"/>
for <font color="Olive">r'</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r'</font>
 
<b>by </b><i><a class="ref" href="complsp1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E4:86"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">X</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r'</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:86_1"/>
<font color="Maroon">r'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:86_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:86_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r'</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T30" target="_self">Th30</a>, <a class="ref" href="complsp1.html#T32" target="_self">Th32</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E5:86_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T31" target="_self">Th31</a>, <a class="txt" href="complsp1.html#E54"><i><font color="Green">Lemma71</font></i></a></i>;<br/><b>hence </b><a NAME="E6:86_1"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T35" target="_self">Th35</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="E6:86"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="complsp1.html#E48"><i><font color="Green">Lemma66</font></i></a></i>;<br/>


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