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<b>let </b><font color="Maroon">a1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:51"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or ( <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>suppose </b></a><a NAME="E1:51_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:51_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E4:51_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:51_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:51_1_1"/>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:51_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E2:51_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E3:51_1_2"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:51_1_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E6:51_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:51_1_2"/>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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