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<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</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#V5" title="XXREAL_2:attr.5">bounded</a> implies <font color="Maroon" title="c1">X</font> is <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a> )</span><br/>

<b>assume </b><a NAME="E1:112_1_1"/><i><font color="Green" title="E69">Z0</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</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="membered.html#V3" title="MEMBERED:attr.3">real-membered</a></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D3" title="MEMBERED:def.3">MEMBERED:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c2">x</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> )</span><br/>

<b>assume </b><a NAME="E2:112_1_1"/><i><font color="Green" title="E70">Z</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">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="c4">X</font> = <font color="Maroon" title="c1">X</font> as  <a href="membered.html#V2" title="MEMBERED:attr.2">ext-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>  <b>by </b><i><a class="txt" href="#E2:112_1_1"><i><font color="Green" title="E70">Z</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">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><br/><a NAME="E6:112_1_1"/><i><font color="Green" title="E71">W1</font></i>: 
<font color="Maroon" title="c5">r</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="xxreal_2.html#D9" target="_self" title="XXREAL_2:def.9">Def9</a>, <a class="txt" href="#E1:112_1_1"><i><font color="Green" title="E69">Z0</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">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="E8:112_1_1"/><i><font color="Green" title="E72">W2</font></i>: 
<font color="Maroon" title="c6">s</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="xxreal_2.html#D10" target="_self" title="XXREAL_2:def.10">Def10</a>, <a class="txt" href="#E1:112_1_1"><i><font color="Green" title="E69">Z0</font></i></a></i>;<br/>
<a NAME="E9:112_1_1"/><i><font color="Green" title="E73">A</font></i>: 
( <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">s</font> )
 
<b>by </b><i><a class="txt" href="#E2:112_1_1"><i><font color="Green" title="E70">Z</font></i></a>, <a class="txt" href="#E6:112_1_1"><i><font color="Green" title="E71">W1</font></i></a>, <a class="txt" href="#E8:112_1_1"><i><font color="Green" title="E72">W2</font></i></a>, <a class="ref" href="xxreal_2.html#D1" target="_self" title="XXREAL_2:def.1">Def1</a>, <a class="ref" href="xxreal_2.html#D2" target="_self" title="XXREAL_2:def.2">Def2</a></i>;<br/>
<a NAME="E10:112_1_1"/>
( <font color="Maroon" title="c5">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="c6">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="E11:112_1_1"/><b>then </b>
<font color="Maroon" title="c3">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="#E9:112_1_1"><i><font color="Green" title="E73">A</font></i></a>, <a class="ref" href="xxreal_0.html#T45" title="XXREAL_0:th.45">XXREAL_0:45</a></i>;<br/>
<b>hence </b><a NAME="E12:112_1_1"/>
<font color="Maroon" title="c2">x</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 ; <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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