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

<b>set </b><font color="Maroon" title="c1">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c2">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <font color="Maroon" title="c4">Y</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1"/>
( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1"/><i><font color="Green" title="E2">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1"/>
(  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  or  not  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_1"/><i><font color="Green" title="E3">A3</font></i>: 
 <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 ;<br/><div class="add"><b>take </b>
 <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_1"/>
for <font color="Olive" title="b1">r</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="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a></i>;<br/><a NAME="E3:1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:1_1_1_1_1_1"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_1_1_1"/>
for <font color="Olive" title="b1">r</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="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_2"/><i><font color="Green" title="E3">A4</font></i>: 
not  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 ;<br/><div class="add"><b>take </b>
 <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_2"/>
for <font color="Olive" title="b1">r</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="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">r</font> be  <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> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_1_1_2_1"/><i><font color="Green" title="E4">A5</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">r</font>]
 ;<br/>

<a NAME="E2:1_1_1_1_1_2_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E3:1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E5">A6</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="xxreal_1.html#E1:1_1_1_1_1_2_1"><i><font color="Green" title="E4">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 </span><b>by </b><i><a class="txt" href="xxreal_1.html#E3:1_1_1_1_1_2_1"><i><font color="Green" title="E5">A6</font></i></a>, <a class="txt" href="xxreal_1.html#E1:1_1_1_1_1_2"><i><font color="Green" title="E3">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:1_1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="xxreal_1.html#E3:1_1_1_1_1_2_1"><i><font color="Green" title="E5">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E3:1_1_1_1_1_2"/>
for <font color="Olive" title="b1">r</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="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2"/><i><font color="Green" title="E2">A7</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_2_1"/>
(  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  or  not  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_1"/><i><font color="Green" title="E3">A8</font></i>: 
 <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 ;<br/><div class="add"><b>take </b>
 <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
;<br/><b>thus </b><a NAME="E2:1_1_1_2_1_1"/>
for <font color="Olive" title="b1">r</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="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] )
 
<b>by </b><i><a class="txt" href="xxreal_1.html#E1:1_1_1_2_1_1"><i><font color="Green" title="E3">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:1_1_1_2_1_1_1"/>
for <font color="Olive" title="b1">r</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="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:1_1_1_2_1_1"/>
for <font color="Olive" title="b1">r</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="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_2"/><i><font color="Green" title="E3">A9</font></i>: 
not  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 ;<br/><div class="add"><b>take </b>
 <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
;<br/><b>thus </b><a NAME="E2:1_1_1_2_1_2"/>
for <font color="Olive" title="b1">r</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="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a></i>;<br/><b>let </b><font color="Maroon" title="c5">r</font> be  <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> ;<br/><b>assume </b><a NAME="E3:1_1_1_2_1_2"/><i><font color="Green" title="E4">A10</font></i>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">r</font>]
 ;<br/><a NAME="E4:1_1_1_2_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E5:1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E5">A11</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="xxreal_1.html#E3:1_1_1_2_1_2"><i><font color="Green" title="E4">A10</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_2_1_2_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 </span><b>by </b><i><a class="txt" href="xxreal_1.html#E5:1_1_1_2_1_2"><i><font color="Green" title="E5">A11</font></i></a>, <a class="txt" href="xxreal_1.html#E1:1_1_1_2_1_2"><i><font color="Green" title="E3">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_2_1_2_1_1"/>
 <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_2_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_2_1_2_1_2"/>
 <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:1_1_1_2"><i><font color="Green" title="E2">A7</font></i></a>, <a class="txt" href="xxreal_1.html#E5:1_1_1_2_1_2"><i><font color="Green" title="E5">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_3"/><i><font color="Green" title="E2">A12</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:1_1_1_3"/><i><font color="Green" title="E3">A13</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E3:1_1_1_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">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="c3">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</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="Olive" title="b2">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E2:1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:1_1_1_3_1"/><i><font color="Green" title="E4">A14</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>and </b><br/><a NAME="E5:1_1_1_3_1"/><i><font color="Green" title="E5">A15</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">a</font>]
 ;<br/>
<b>assume </b><a NAME="E6:1_1_1_3_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ;<br/>

<a NAME="E7:1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:1_1_1_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] )
 
;<br/>
<b>hence </b><a NAME="E9:1_1_1_3_1"/>
<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">y</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E4:1_1_1_3_1"><i><font color="Green" title="E4">A14</font></i></a>, <a class="txt" href="xxreal_1.html#E5:1_1_1_3_1"><i><font color="Green" title="E5">A15</font></i></a>, <a class="txt" href="xxreal_1.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c5">s</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:1_1_1_3"/><i><font color="Green" title="E4">A16</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">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="c3">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</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="c5">s</font> &amp; <font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> )
 <b>by </b><i><a class="ref" href="axioms.html#T26" title="AXIOMS:th.26">AXIOMS:26</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">s</font> as  <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>  ;<br/><b>take </b>
<font color="Maroon" title="c6">s</font>
;<br/><b>thus </b><a NAME="E7:1_1_1_3"/>
for <font color="Olive" title="b1">r</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="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">r</font> be  <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> ;<br/>

<b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:1_1_1_3_2"/><i><font color="Green" title="E5">A17</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E2:1_1_1_3"><i><font color="Green" title="E3">A13</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:1_1_1_3_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="xxreal_1.html#E2:1_1_1_3_2"><i><font color="Green" title="E5">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:1_1_1_3_2"/><b>then </b><i><font color="Green" title="E6">A18</font></i>: 
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] )
 
;<br/>
<b>assume </b><a NAME="E5:1_1_1_3_2"/><i><font color="Green" title="E7">A19</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">r</font>]
 ;<br/>

<a NAME="E6:1_1_1_3_2"/><b>then </b><i><font color="Green" title="E8">A20</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="xxreal_1.html#E4:1_1_1_3_2"><i><font color="Green" title="E6">A18</font></i></a>, <a class="txt" href="xxreal_1.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_3_2_1"/>
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 </span><b>by </b><i><a class="txt" href="xxreal_1.html#E6:1_1_1_3_2"><i><font color="Green" title="E8">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T13" title="XXREAL_0:th.13">XXREAL_0:13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_3_2_1_1"/><i><font color="Green" title="E9">A21</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c7">r</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E3:1_1_1_3_2_1_1"/>
<font color="Maroon" title="c9">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E4:1_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="xxreal_1.html#E5:1_1_1_3_2"><i><font color="Green" title="E7">A19</font></i></a></i>;<br/><a NAME="E5:1_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:1_1_1_3_2_1_1"><i><font color="Green" title="E9">A21</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_3_2_1_1"/>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E2:1_1_1_3_2"><i><font color="Green" title="E5">A17</font></i></a>, <a class="txt" href="xxreal_1.html#E5:1_1_1_3"><i><font color="Green" title="E4">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_3_2_1_2"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_3_2_1_2"/>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>let </b><font color="Maroon" title="c7">r</font> be  <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> ;<br/><b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E9:1_1_1_3"/><i><font color="Green" title="E5">A22</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:1_1_1_3"><i><font color="Green" title="E2">A12</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E10:1_1_1_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="xxreal_1.html#E9:1_1_1_3"><i><font color="Green" title="E5">A22</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:1_1_1_3"/><b>then </b><i><font color="Green" title="E6">A23</font></i>: 
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )
 ;<br/><b>assume </b><a NAME="E12:1_1_1_3"/><i><font color="Green" title="E7">A24</font></i>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">r</font>]
 ;<br/><a NAME="E13:1_1_1_3"/><b>then </b><i><font color="Green" title="E8">A25</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E11:1_1_1_3"><i><font color="Green" title="E6">A23</font></i></a>, <a class="txt" href="xxreal_1.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_3_3"/>
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="xxreal_1.html#E13:1_1_1_3"><i><font color="Green" title="E8">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T10" title="XXREAL_0:th.10">XXREAL_0:10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_3_1"><b>suppose </b></a><a NAME="E1:1_1_1_3_3_1"/><i><font color="Green" title="E9">A26</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c7">r</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E3:1_1_1_3_3_1"/>
<font color="Maroon" title="c9">r</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E4:1_1_1_3_3_1"/><b>then </b>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="xxreal_1.html#E12:1_1_1_3"><i><font color="Green" title="E7">A24</font></i></a></i>;<br/><a NAME="E5:1_1_1_3_3_1"/><b>then </b>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:1_1_1_3_3_1"><i><font color="Green" title="E9">A26</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_3_3_1"/>
<font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E9:1_1_1_3"><i><font color="Green" title="E5">A22</font></i></a>, <a class="txt" href="xxreal_1.html#E5:1_1_1_3"><i><font color="Green" title="E4">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_3_2"><b>suppose </b></a><a NAME="E1:1_1_1_3_3_2"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_3_3_2"/>
<font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
