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<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#V3" title="MEMBERED:attr.3">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">r</font> =  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>let </b><font color="Maroon" title="c4">t</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E38">A1</font></i>: 
for <font color="Olive" title="b1">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>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font>
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D1" target="_self" title="SEQ_4:def.1">Def1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">s</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">t</font>;<br/>
<b>assume </b><a NAME="E3:53"/>
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">t</font>
 ;<br/>

<a NAME="E4:53"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">t'</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="E6:53"/><i><font color="Green" title="E40">A3</font></i>: 
( <font color="Maroon" title="c6">t'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">t'</font> )
 <b>by </b><i><a class="txt" href="seq_4.html#E2:53"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="seq_4.html#D4" target="_self" title="SEQ_4:def.4">Def4</a></i>;<br/>
<b>thus </b><a NAME="E7:53"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_4.html#E1:53"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="seq_4.html#E6:53"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>


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