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<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>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>assume </b><b>that </b><br/><a NAME="E1:52"/><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#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st ( 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#NR2" target="_self">&gt;=</a> <font color="Olive">t</font> ) holds <br/><font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">t</font>
 ;<br/>

<a NAME="E3:52"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> is <a href="seq_4.html#V2" target="_self">bounded_below</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</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="E1:52_1"/><i><font color="Green">E49</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><b>assume </b><a NAME="E2:52_1"/>
for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E3:52_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>hence </b><a NAME="E4:52_1"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_4.html#E4"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:52"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K3" target="_self">lower_bound</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/>


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