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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">X</font> is <a href="supinf_1.html#V4" target="_self">bounded_below</a> &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon">LB0</font> being    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:41"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">LB0</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:41"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">LB0</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:41"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:41"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 <b>by </b><i>, <a class="txt" href="supinf_1.html#E4:41"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <a href="supinf_1.html#K3" target="_self">REAL</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3" target="_self">REAL</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <a href="supinf_1.html#K3" target="_self">REAL</a>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font></span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3" target="_self">REAL</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">LB0</font> = <font color="Maroon">LB0</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="supinf_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E10:41"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T22" target="_self">Th22</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/><a NAME="E2:41_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:41_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:41_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">a</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">b</font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">b</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><a NAME="E6:41_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">d</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="supinf_1.html#E1:41"><i><font color="Green">E22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">a</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><a NAME="E8:41_1"/>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T29" target="_self">Th29</a>, </i>;<br/><a NAME="E9:41_1"/><b>then </b>
 ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:41_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">LB</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E13:41"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b</font>, <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/>( <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> )
 <b>by </b><i><a class="ref" href="axioms.html#T26">AXIOMS:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">LB</font> = <font color="Maroon">LB</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">d</font> = <font color="Maroon">LB</font>;<br/>
<b>reconsider </b><font color="Maroon">LB</font> = <font color="Maroon">LB</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E16:41"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b</font>, <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/>( <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">a</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E2:41_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  implies ( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 ;<br/>

<a NAME="E2:41_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_2_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">LB0</font> = <font color="Maroon">LB0</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E3:41_2_1_1_1"/>
<font color="Maroon">LB0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:41_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_1.html#D9" target="_self">Def9</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:41_2_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> ,<a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:41_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T29" target="_self">Th29</a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:41_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:41_2_1"/>
( <font color="Maroon">LB0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:41_2_1_2"/>
<font color="Maroon">LB0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:41_2_1_2"/>
( <font color="Maroon">LB0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:41_2_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="supinf_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:41_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  implies ( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_2"/><i><font color="Green">E81</font></i>: 
( not <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 ;<br/>

<a NAME="E2:41_2_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">b</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="txt" href="supinf_1.html#E1:41"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E4:41_2_2"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T22" target="_self">Th22</a>, <a class="ref" href="supinf_1.html#D9" target="_self">Def9</a>, </i>;<br/>
<a NAME="E5:41_2_2"/><b>then </b><i><font color="Green">E83</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:41_2_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> ,<a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="supinf_1.html#T33" target="_self">Th33</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E8:41_2_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E10:41_2_2"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="ref" href="supinf_1.html#T34" target="_self">Th34</a></i>;<br/>
<b>hence </b><a NAME="E11:41_2_2"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:41_2"/>
( not <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; not <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  implies ( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_3"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 ;<br/>

<a NAME="E2:41_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_2_3_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_3_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">LB0</font> = <font color="Maroon">LB0</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E3:41_2_3_1_1"/>
<font color="Maroon">LB0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:41_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_1.html#T33" target="_self">Th33</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:41_2_3_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> ,<a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:41_2_3_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D9" target="_self">Def9</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:41_2_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">b</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="txt" href="supinf_1.html#E1:41"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E6:41_2_3"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T22" target="_self">Th22</a>, <a class="ref" href="supinf_1.html#D9" target="_self">Def9</a>, </i>;<br/>
<a NAME="E7:41_2_3"/><b>then </b><i><font color="Green">E89</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E8:41_2_3"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> ,<a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="supinf_1.html#T33" target="_self">Th33</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E10:41_2_3"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:41_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:41_2_3"/><b>then </b>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T34" target="_self">Th34</a></i>;<br/>
<b>hence </b><a NAME="E13:41_2_3"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:41_2"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font> &amp; <font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T3">XXREAL_0:3</a>, <a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">LB</font>
;<br/>

<a NAME="E17:41"/>
for <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T18" target="_self">Th18</a>, </i>;<br/>
<b>hence </b><a NAME="E18:41"/>
<font color="Maroon">LB</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">y</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E19:41"/>
<font color="Maroon">y</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font>
 ;<br/>

<a NAME="E20:41"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:41"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:41"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T22" target="_self">Th22</a>, </i>;<br/>


</div>
