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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="membered.html#V2">real-membered</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>set </b><font color="Maroon">r</font> =  <a href="seq_4.html#K2" target="_self">upper_bound</a> <font color="Maroon">X</font>;<br/>
<b>let </b><font color="Maroon">t</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:53"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">X</font> is <a href="seq_4.html#V1" target="_self">bounded_above</a>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K2" target="_self">upper_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t</font>;<br/>
<b>assume </b><a NAME="E3:53"/>
 <a href="seq_4.html#K2" target="_self">upper_bound</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E4:53"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K2" target="_self">upper_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<b>then consider </b><font color="Maroon">t'</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:53"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><a href="seq_4.html#K2" target="_self">upper_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" target="_self">upper_bound</a> <font color="Maroon">X</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t'</font> )
 <b>by </b><i><a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E7:53"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="seq_4.html#E4"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


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