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<b>let </b><font color="Maroon" title="c1">eps</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/><b>let </b><font color="Maroon" title="c2">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="E1:29"/><i><font color="Green" title="E19">A1</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> &amp;  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 ;<br/>

<b>assume </b><a NAME="E2:29"/>
for <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> or  not <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">eps</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> )
 ;<br/>

<a NAME="E3:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">eps</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a></i>;<br/>
<a NAME="E4:29"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">eps</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" title="SUPINF_1:def.16">SUPINF_1:def 16</a></i>;<br/>
<a NAME="E5:29"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c1">eps</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1"/>
( <font color="Maroon" title="c1">eps</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or <font color="Maroon" title="c1">eps</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="measure6.html#E5:29"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>suppose </b></a><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c1">eps</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">a</font> =  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c1">eps</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="measure6.html#E1:29"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="measure6.html#T15" target="_self" title="MEASURE6:th.15">Th15</a></i>;<br/><a NAME="E3:29_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">eps</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/><a NAME="E4:29_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="measure6.html#E4:29"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/><a NAME="E5:29_1_1"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/><b>hence </b><a NAME="E6:29_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E1:29"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><b>suppose </b></a><a NAME="E1:29_1_2"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c1">eps</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/><div class="add"><a NAME="E2:29_1_2"/><i><font color="Green" title="E23">A5</font></i>: 
( not  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</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  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E1:29"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><a NAME="E3:29_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">eps</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="measure6.html#E1:29_1_2"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="supinf_2.html#T6" title="SUPINF_2:th.6">SUPINF_2:6</a></i>;<br/><b>hence </b><a NAME="E4:29_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E4:29"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="measure6.html#E2:29_1_2"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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