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<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a> non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> implies  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<b>given </b><font color="Maroon" title="c2">r</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><a NAME="E2:38"/><i><font color="Green" title="E30">G</font></i>: 
<font color="Maroon" title="c2">r</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of <font color="Maroon" title="c1">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xxreal_2.html#D10" target="_self" title="XXREAL_2:def.10">XXREAL_2:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/>

<a NAME="E3:38"/><i><font color="Green" title="E31">A</font></i>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="ref" href="xxreal_2.html#D3" target="_self" title="XXREAL_2:def.3">Def3</a>, <a class="txt" href="#E2:38"><i><font color="Green" title="E30">G</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">s</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="E5:38"/><i><font color="Green" title="E32">W</font></i>: 
<font color="Maroon" title="c3">s</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="membered.html#T9" title="MEMBERED:th.9">MEMBERED:9</a></i>;<br/>
<a NAME="E6:38"/><i><font color="Green" title="E33">F</font></i>: 
( <font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.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/>
<a NAME="E7:38"/>
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E32">W</font></i></a>, <a class="ref" href="xxreal_2.html#T4" target="_self" title="XXREAL_2:th.4">Th3B</a></i>;<br/>
<b>hence </b><a NAME="E8:38"/>
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="#E3:38"><i><font color="Green" title="E31">A</font></i></a>, <a class="txt" href="#E6:38"><i><font color="Green" title="E33">F</font></i></a>, <a class="ref" href="xxreal_0.html#T45" title="XXREAL_0:th.45">XXREAL_0:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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