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<div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">b</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">c</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E2:25"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> or <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
;<br/>
<a NAME="E3:25"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T96">SUPINF_1:96</a></i>;<br/>
<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="E5:25"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">a</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>  <a href="supinf_1.html#K7">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span> &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/>;<br/>
<a NAME="E6:25"/>
 ex <font color="Olive">o</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Olive">o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E1:25_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:25"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T98">SUPINF_1:98</a></i>;<br/>
<a NAME="E8:25"/>
 ex <font color="Olive">o</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Olive">o</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>thus </b><a NAME="E1:25_2"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:25"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T98">SUPINF_1:98</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E10:25"/>
( <font color="Maroon">a</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">b</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</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>, , <a class="ref" href="measure5.html#T1" target="_self">Th1</a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>


</div>
