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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">B</font> = <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <b>by </b><i><a class="txt" href="urysohn2.html#E1:27"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="urysohn2.html#E3:27"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="numbers.html#T31" title="NUMBERS:th.31">NUMBERS:31</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">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" title="b1">x</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E2:27_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E3:27_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:27"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="supinf_1.html#T39" title="SUPINF_1:th.39">SUPINF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:27_1"/>
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D17" title="SUPINF_1:def.17">SUPINF_1:def 17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:27"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">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" title="b1">x</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E2:27_2"/><b>then </b>
<font color="Maroon" title="c3">b</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a></i>;<br/>
<a NAME="E3:27_2"/><b>then </b>
<font color="Maroon" title="c3">b</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:27"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="supinf_1.html#T34" title="SUPINF_1:th.34">SUPINF_1:34</a></i>;<br/>
<b>hence </b><a NAME="E4:27_2"/>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16" title="SUPINF_1:def.16">SUPINF_1:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:27"/>
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:27"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>


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