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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V3" title="SUPINF_1:attr.3">bounded_above</a> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">UB0</font> being    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:40"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c2">UB0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#D11" target="_self" title="SUPINF_1:def.11">Def11</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c2">UB0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D14" target="_self" title="SUPINF_1:def.14">Def14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:40"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:40"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="supinf_1.html#T59" target="_self" title="SUPINF_1:th.59">Th59</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">S</font> = <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">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/>
<b>reconsider </b><font color="Maroon" title="c5">Y</font> = <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">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/>
<b>reconsider </b><font color="Maroon" title="c6">UB0</font> = <font color="Maroon" title="c2">UB0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="supinf_1.html#E3:40"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>
<a NAME="E10:40"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E6:40"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</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="E1:40_1"/><i><font color="Green" title="E31">A6</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 ;<br/><a NAME="E2:40_1"/><b>then </b><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <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="E3:40_1"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_1"><i><font color="Green" title="E31">A6</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="E4:40_1"/><i><font color="Green" title="E34">A9</font></i>: 
( <font color="Maroon" title="c7">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp; <font color="Maroon" title="c8">b</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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/><b>reconsider </b><font color="Maroon" title="c9">d</font> = <font color="Maroon" title="c8">b</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</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="E6:40_1"/><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c9">d</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E3:40_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="supinf_1.html#D14" target="_self" title="SUPINF_1:def.14">Def14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c7">a</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</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="E8:40_1"/>
<font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E2:40_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="supinf_1.html#E6:40_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="supinf_1.html#D9" target="_self" title="SUPINF_1:def.9">Def9</a></i>;<br/><a NAME="E9:40_1"/><b>then </b>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">q</font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E4:40_1"><i><font color="Green" title="E34">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E10:40_1"/>
<font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font>
 ;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c7">UB</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="E13:40"/><i><font color="Green" title="E31">A11</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">UB</font> &amp; <font color="Maroon" title="c7">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> )
 <b>by </b><i><a class="ref" href="axioms.html#T26" title="AXIOMS:th.26">AXIOMS:26</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">d</font> = <font color="Maroon" title="c7">UB</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">UB</font> = <font color="Maroon" title="c7">UB</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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/>
<b>reconsider </b><font color="Maroon" title="c10">UB</font> = <font color="Maroon" title="c9">UB</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</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/>
<div><i><font color="Green" title="E32">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/><b>assume </b><a NAME="E1:40_2"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font> )
 ;<br/><b>thus </b><a NAME="E2:40_2"/>
( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">UB</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_1"/><i><font color="Green" title="E34">A14</font></i>: 
( not <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  implies ( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">UB</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2_1_1"/><i><font color="Green" title="E35">A15</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ;<br/>

<a NAME="E2:40_2_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_1_1_1"/><i><font color="Green" title="E36">A16</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2_1_1_1_1"/><i><font color="Green" title="E37">A17</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">UB0</font> = <font color="Maroon" title="c6">UB0</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<a NAME="E3:40_2_1_1_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">UB0</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="supinf_1.html#D9" target="_self" title="SUPINF_1:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E4:40_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2_1_1_1_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="txt" href="supinf_1.html#E25"><i><font color="Green" title="E5">Lm2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40_2_1_1_1"/>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:40_2_1_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2_1_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="supinf_1.html#E1:40_2_1_1_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E4:40_2_1_1"/><i><font color="Green" title="E36">A18</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2"><i><font color="Green" title="E33">A13</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="E5:40_2_1_1"/>
( <font color="Maroon" title="c13">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">UB</font> &amp; <font color="Maroon" title="c7">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">UB0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_1_1_2"/>
<font color="Maroon" title="c6">UB0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E4:40"><i><font color="Green" title="E28">A3</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="E2:40_2_1_1_2"/>
( <font color="Maroon" title="c13">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">UB</font> &amp; <font color="Maroon" title="c7">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">UB0</font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E13:40"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="supinf_1.html#E4:40_2_1_1"><i><font color="Green" title="E36">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:40_2_1_1"/>
( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">UB</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2_1_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40_2_1"/><i><font color="Green" title="E35">A19</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; not <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  implies ( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">UB</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2_1_2"/><i><font color="Green" title="E36">A20</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ;<br/>

<a NAME="E2:40_2_1_2"/><i><font color="Green" title="E37">A21</font></i>: 
<font color="Maroon" title="c12">b</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="supinf_1.html#D14" target="_self" title="SUPINF_1:def.14">Def14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E4:40_2_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E6:40"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="supinf_1.html#E2:40_2_1_2"><i><font color="Green" title="E37">A21</font></i></a>, <a class="ref" href="supinf_1.html#D9" target="_self" title="SUPINF_1:def.9">Def9</a></i>;<br/>
<a NAME="E5:40_2_1_2"/><b>then </b><i><font color="Green" title="E38">A22</font></i>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_1.html#E24"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/>
<a NAME="E6:40_2_1_2"/>
( <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  or <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c12">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2_1_2"><i><font color="Green" title="E36">A20</font></i></a>, <a class="txt" href="supinf_1.html#E5:40_2_1_2"><i><font color="Green" title="E38">A22</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E8:40_2_1_2"/><i><font color="Green" title="E39">A23</font></i>: 
<font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c13">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E10:40_2_1_2"/>
( <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">UB</font> &amp; <font color="Maroon" title="c7">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font> )
 
<b>by </b><i><a class="txt" href="supinf_1.html#E10:40"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="supinf_1.html#E13:40"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="supinf_1.html#E8:40_2_1_2"><i><font color="Green" title="E39">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:40_2_1_2"/>
( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">UB</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2_1_2"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:40_2_1"/>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  implies ( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">UB</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2_1_3"/><i><font color="Green" title="E36">A24</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ;<br/>

<a NAME="E2:40_2_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_1_3_1"/><i><font color="Green" title="E37">A25</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2_1_3_1_1"/><i><font color="Green" title="E38">A26</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">UB0</font> = <font color="Maroon" title="c6">UB0</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<a NAME="E3:40_2_1_3_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">UB0</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="supinf_1.html#D9" target="_self" title="SUPINF_1:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E4:40_2_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2_1_3_1_1"><i><font color="Green" title="E38">A26</font></i></a>, <a class="txt" href="supinf_1.html#E25"><i><font color="Green" title="E5">Lm2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40_2_1_3_1"/>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:40_2_1_3_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2_1_3"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="supinf_1.html#E1:40_2_1_3_1"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E4:40_2_1_3"/><i><font color="Green" title="E37">A27</font></i>: 
<font color="Maroon" title="c12">b</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="supinf_1.html#D14" target="_self" title="SUPINF_1:def.14">Def14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E6:40_2_1_3"/>
<font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E6:40"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="supinf_1.html#E4:40_2_1_3"><i><font color="Green" title="E37">A27</font></i></a>, <a class="ref" href="supinf_1.html#D9" target="_self" title="SUPINF_1:def.9">Def9</a></i>;<br/>
<a NAME="E7:40_2_1_3"/><b>then </b><i><font color="Green" title="E38">A28</font></i>: 
not <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_1.html#E24"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/>
<a NAME="E8:40_2_1_3"/>
( <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  or <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c12">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2_1_3"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="supinf_1.html#E7:40_2_1_3"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E10:40_2_1_3"/><i><font color="Green" title="E39">A29</font></i>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2"><i><font color="Green" title="E33">A13</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="E11:40_2_1_3"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2"><i><font color="Green" title="E33">A13</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="E12:40_2_1_3"/><b>then </b>
( <font color="Maroon" title="c13">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">UB</font> &amp; <font color="Maroon" title="c7">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font> )
 
<b>by </b><i><a class="txt" href="supinf_1.html#E13:40"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="supinf_1.html#E10:40_2_1_3"><i><font color="Green" title="E39">A29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:40_2_1_3"/>
( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">UB</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_2_1"/>
( <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">UB</font> &amp; <font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">b</font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:40_2_1"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="supinf_1.html#E2:40_2_1"><i><font color="Green" title="E35">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a>, <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>
<a NAME="E17:40"/><i><font color="Green" title="E33">A30</font></i>: 
<font color="Maroon" title="c10">UB</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>proof </b></a><div class="add">

<a NAME="E1:40_3"/>
for <font color="Olive" title="b1">a</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">UB</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E4:40"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="supinf_1.html#E16:40"><i><font color="Green" title="E32">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:40_3"/>
<font color="Maroon" title="c10">UB</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9" target="_self" title="SUPINF_1:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:40"/><i><font color="Green" title="E34">A31</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b1">y</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:40_4"/>
<font color="Maroon" title="c11">y</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:40_4"/><b>then </b>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D14" target="_self" title="SUPINF_1:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E3:40_4"/>
<font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E6:40"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="supinf_1.html#E16:40"><i><font color="Green" title="E32">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c10">UB</font>
;<br/>

<b>thus </b><a NAME="E19:40"/>
( <font color="Maroon" title="c10">UB</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b1">y</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c10">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y</font> ) )
 <b>by </b><i><a class="txt" href="supinf_1.html#E17:40"><i><font color="Green" title="E33">A30</font></i></a>, <a class="txt" href="supinf_1.html#E18:40"><i><font color="Green" title="E34">A31</font></i></a></i>;<br/>


</div>
