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<b>let </b><font color="Maroon">eps</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><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="E1:28"/><i><font color="Green">E18</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">eps</font> &amp;  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font> is   <a href="real_1.html#NM1">Real</a> )
 ;<br/>

<b>assume </b><a NAME="E2:28"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">eps</font> )
 ;<br/>

<a NAME="E3:28"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">eps</font> is    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E4:28"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">eps</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>
<a NAME="E5:28"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">eps</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( <font color="Maroon">eps</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">eps</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/>
<font color="Maroon">eps</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a</font> =  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font>, <font color="Maroon">b</font> = <font color="Maroon">eps</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="measure6.html#E4:28"><i><font color="Green">E19</font></i></a>, </i>;<br/><a NAME="E3:28_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">eps</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E4:28_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E5:28_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:28_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E7:28_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E4:28"><i><font color="Green">E19</font></i></a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">eps</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:28_1_2"/><i><font color="Green">E26</font></i>: 
( not  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E4:28"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E3:28_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">eps</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure6.html#E5:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:28_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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