<?xml version="1.0"?>
<div class="add">

<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">m</font>, <font color="Maroon" title="c3">M</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>

<b>assume </b><a NAME="E2:40"/><i><font color="Green" title="E29">A2</font></i>: 
( ( for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>for <font color="Olive" title="b3">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">e</font> &amp; <font color="Olive" title="b3">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">d</font> holds <br/><font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; not <font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 ;<br/>

<b>thus </b><a NAME="E3:40"/>
<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#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">M</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<a NAME="E3:40_1"/>
( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="measure6.html#E1:40_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="supinf_1.html#T76" title="SUPINF_1:th.76">SUPINF_1:76</a>, <a class="ref" href="supinf_1.html#T79" title="SUPINF_1:th.79">SUPINF_1:79</a></i>;<br/>
<a NAME="E4:40_1"/><b>then </b>
( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="measure6.html#E1:40_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E5:40_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">M</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="measure6.html#E1:40_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="measure5.html#D4" title="MEASURE5:def.4">MEASURE5:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:40"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">M</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 ;<br/>

<a NAME="E5:40"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="numbers.html#T31" title="NUMBERS:th.31">NUMBERS:31</a></i>;<br/>
<a NAME="E7:40"/><i><font color="Green" title="E31">A5</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="measure6.html#E4:40"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="measure5.html#D4" title="MEASURE5:def.4">MEASURE5:def 4</a></i>;<br/>
<a NAME="E8:40"/>
 ex <font color="Olive" title="b1">b</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 <br/>( <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2"/>
for <font color="Olive" title="b1">b</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>   holds <br/>(  not <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font> or  not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>

<a NAME="E2:40_2"/><b>then </b>
for <font color="Olive" title="b1">a</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font>
 
;<br/>
<a NAME="E3:40_2"/><b>then </b>
<font color="Maroon" title="c5">x</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="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:40_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="measure6.html#E7:40"><i><font color="Green" title="E31">A5</font></i></a>, <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>then consider </b><font color="Maroon" title="c6">b</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E10:40"/><i><font color="Green" title="E32">A6</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>
<b>thus </b><a NAME="E11:40"/>
<font color="Maroon" title="c4">x</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="txt" href="measure6.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="measure6.html#E7:40"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="measure6.html#E10:40"><i><font color="Green" title="E32">A6</font></i></a></i>;<br/>


</div>
