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

<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
<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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <br/>( (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> ) &amp; (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> ) )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
 <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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <br/>( (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> ) &amp; (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> ) )</span><br/><div class="add"><span class="kw">take </span>
 <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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>  ) &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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> ) )</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_1_1"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" onmouseover="rs('xxreal_0/T3')" onmouseout="rh()">XXREAL_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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></span><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>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_1"><span class="lab"><font color="Green" title="E3">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_1_1_1"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E3">A4</font></span>: 
 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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <br/>( (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> ) &amp; (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> ) )</span><br/><div class="add"><span class="kw">take </span>
 <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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>  ) &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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> ) )</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_1_2"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">r</font>] implies <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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E4">A5</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">r</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E3:1_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A6</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E4">A5</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_2"><span class="lab"><font color="Green" title="E3">A4</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T14" onmouseover="rs('xxreal_0/T14')" onmouseout="rh()">XXREAL_0:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><div class="add"><span class="kw">hence </span><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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><div class="add"><span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_2"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" onmouseover="rs('xxreal_0/T5')" onmouseout="rh()">XXREAL_0:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_2"/><span class="lab"><font color="Green" title="E2">A7</font></span>: 
<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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <br/>( (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> ) &amp; (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> ) )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E3">A8</font></span>: 
 <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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <br/>( (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> ) &amp; (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> ) )</span><br/><div class="add"><span class="kw">take </span>
 <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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>  ) &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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> ) )</span><br/><a NAME="E2:1_1_1_2_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>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_2_1_1"><span class="lab"><font color="Green" title="E3">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_1_2_1_1"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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></span><br/><span class="kw">thus </span><a NAME="E4:1_1_1_2_1_1"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" onmouseover="rs('xxreal_0/T5')" onmouseout="rh()">XXREAL_0:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E3">A9</font></span>: 
 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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <br/>( (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> ) &amp; (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> ) )</span><br/><div class="add"><span class="kw">take </span>
 <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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>  ) &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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> ) )</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_2_1_2"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" onmouseover="rs('xxreal_0/T3')" onmouseout="rh()">XXREAL_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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></span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">r</font>] implies  <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> )</span><br/><span class="kw">assume </span><a NAME="E3:1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E4">A10</font></span>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">r</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><a NAME="E5:1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A11</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_2_1_2"><span class="lab"><font color="Green" title="E4">A10</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_2_1_2"><span class="lab"><font color="Green" title="E3">A9</font></span></a>, <a class="txt" href="#E5:1_1_1_2_1_2"><span class="lab"><font color="Green" title="E5">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T14" onmouseover="rs('xxreal_0/T14')" onmouseout="rh()">XXREAL_0:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2_1_1"><span class="kw">suppose </span></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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">hence </span><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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2_1_2"><span class="kw">suppose </span></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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/><div class="add"><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_2"><span class="lab"><font color="Green" title="E2">A7</font></span></a>, <a class="txt" href="#E5:1_1_1_2_1_2"><span class="lab"><font color="Green" title="E5">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_3"/><span class="lab"><font color="Green" title="E2">A12</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_3"/><span class="lab"><font color="Green" title="E3">A13</font></span>: 
<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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> st <br/>( (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> ) &amp; (  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> ) )</span><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#NM1" title="XREAL_0:NM.1">Real</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> implies <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> )</span><br/>

<span class="kw">assume </span><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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<a NAME="E2:1_1_1_3_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A14</font></span>: 
 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="c5">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )
 
;<br/>
<span class="kw">assume </span><a NAME="E4:1_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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E5:1_1_1_3_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:1_1_1_3_1"/><span class="kw">then </span>
 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/>
<span class="kw">hence </span><a NAME="E7:1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E3:1_1_1_3_1"><span class="lab"><font color="Green" title="E4">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:1_1_1_3"/><span class="lab"><font color="Green" title="E4">A15</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="axioms.html#T1" onmouseover="rs('axioms/T1')" onmouseout="rh()">AXIOMS:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">s</font> as   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">s</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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> ) &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> ) )</span><br/><span class="kw">thus </span><a NAME="E7:1_1_1_3"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="1_1_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">r</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">r</font>] implies <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> )</span><br/>

<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E2:1_1_1_3_2"/><span class="lab"><font color="Green" title="E5">A16</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_3"><span class="lab"><font color="Green" title="E3">A13</font></span></a></span>;<br/>
<a NAME="E3:1_1_1_3_2"/>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI: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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_3_2"><span class="lab"><font color="Green" title="E5">A16</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:1_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A17</font></span>: 
 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/>
<span class="kw">assume </span><a NAME="E5:1_1_1_3_2"/><span class="lab"><font color="Green" title="E7">A18</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">r</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E4:1_1_1_3_2"><span class="lab"><font color="Green" title="E6">A17</font></span></a>, <a class="txt" href="#E5:1_1_1_3_2"><span class="lab"><font color="Green" title="E7">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T13" onmouseover="rs('xxreal_0/T13')" onmouseout="rh()">XXREAL_0:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_3_2_1_1"/><span class="lab"><font color="Green" title="E8">A19</font></span>: 
<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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c7">r</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:1_1_1_3_2_1_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_3_2"><span class="lab"><font color="Green" title="E7">A18</font></span></a></span>;<br/><a NAME="E5:1_1_1_3_2_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_3_2_1_1"><span class="lab"><font color="Green" title="E8">A19</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_3"><span class="lab"><font color="Green" title="E4">A15</font></span></a>, <a class="txt" href="#E2:1_1_1_3_2"><span class="lab"><font color="Green" title="E5">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_2_1_2"><span class="kw">suppose </span></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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" onmouseover="rs('xxreal_0/T5')" onmouseout="rh()">XXREAL_0:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c7">r</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">r</font>] implies <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> )</span><br/><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E9:1_1_1_3"/><span class="lab"><font color="Green" title="E5">A20</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_3"><span class="lab"><font color="Green" title="E2">A12</font></span></a></span>;<br/><a NAME="E10:1_1_1_3"/>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI: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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_3"><span class="lab"><font color="Green" title="E5">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E11:1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A21</font></span>: 
 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/><span class="kw">assume </span><a NAME="E12:1_1_1_3"/><span class="lab"><font color="Green" title="E7">A22</font></span>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">r</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E11:1_1_1_3"><span class="lab"><font color="Green" title="E6">A21</font></span></a>, <a class="txt" href="#E12:1_1_1_3"><span class="lab"><font color="Green" title="E7">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T10" onmouseover="rs('xxreal_0/T10')" onmouseout="rh()">XXREAL_0:10</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_3_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_3_3_1"/><span class="lab"><font color="Green" title="E8">A23</font></span>: 
<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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c7">r</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:1_1_1_3_3_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_3"><span class="lab"><font color="Green" title="E7">A22</font></span></a></span>;<br/><a NAME="E5:1_1_1_3_3_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_3_3_1"><span class="lab"><font color="Green" title="E8">A23</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_3"><span class="lab"><font color="Green" title="E4">A15</font></span></a>, <a class="txt" href="#E9:1_1_1_3"><span class="lab"><font color="Green" title="E5">A20</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_3_3_2"><span class="kw">suppose </span></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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" onmouseover="rs('xxreal_0/T3')" onmouseout="rh()">XXREAL_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
