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<a NAME="E1:20"/>
for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  holds <br/> not <font color="Olive" title="b1">X</font> is <a href="supinf_1.html#V3" title="SUPINF_1:attr.3">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><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="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 ;<br/>

<b>assume </b><a NAME="E2:20_1"/>
<font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V3" title="SUPINF_1:attr.3">bounded_above</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">UB</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="E4:20_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c2">UB</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/>
<b>reconsider </b><font color="Maroon" title="c3">UB</font> = <font color="Maroon" title="c2">UB</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#E4:20_1"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">x</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="E7:20_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c3">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T3" title="XREAL_1:th.3">XREAL_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</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="c6">x</font> = <font color="Maroon" title="c5">x</font> as  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">UB</font> = <font color="Maroon" title="c3">UB</font> as  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E11:20_1"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">UB</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:20_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="supinf_1.html#D9" target="_self" title="SUPINF_1:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>
<b>thus </b><a NAME="E13:20_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_1.html#E7:20_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="supinf_1.html#E11:20_1"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20"/>
not  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  is <a href="supinf_1.html#V3" title="SUPINF_1:attr.3">bounded_above</a>
 ;<br/>


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