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<div class="add">

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">b</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="measure5.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="txt" href="urysohn2.html#E3:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn2.html#E5:27"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E2:27_1"/><b>then </b>
<font color="Maroon">a</font> is    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E3:27_1"/><b>then </b>
<font color="Maroon">a</font> is    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E3:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="supinf_1.html#T39">SUPINF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:27_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:27"/>
 <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_2"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E2:27_2"/><b>then </b>
<font color="Maroon">b</font> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<a NAME="E3:27_2"/><b>then </b>
<font color="Maroon">b</font> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E3:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="supinf_1.html#T34">SUPINF_1:34</a></i>;<br/>
<b>hence </b><a NAME="E4:27_2"/>
 <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:27"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">A</font> &amp;  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>


</div>
