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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>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> is <a href="rcomp_1.html#V1" target="_self">compact</a>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>take </b>
0
;<br/><b>thus </b><a NAME="E2:34_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default">1,0</span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_2"/>
( <font color="Maroon">X</font> is <a href="seq_4.html#V3">bounded</a> &amp; <font color="Maroon">X</font> is <a href="rcomp_1.html#V2" target="_self">closed</a> )
 <b>by </b><i>, , </i>;<br/><a NAME="E3:34_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">X</font> is <a href="seq_4.html#V2">bounded_below</a> &amp; <font color="Maroon">X</font> is <a href="seq_4.html#V1">bounded_above</a> &amp; <font color="Maroon">X</font> is <a href="rcomp_1.html#V2" target="_self">closed</a> )
 <b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/><a NAME="E4:34_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:34_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><b>take </b><font color="Maroon">p</font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font>;<br/><b>take </b><font color="Maroon">g</font> =  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">X</font>;<br/><a NAME="E6:34_1_2"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:34_1_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:34_1_2_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><a NAME="E3:34_1_2_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E4:34_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:34_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 ;<br/></div><b>end;</b></div><a NAME="E8:34_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/><b>hence </b><a NAME="E9:34_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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