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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:59"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="supinf_1.html#V3" target="_self">bounded_above</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> </span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:59"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E4:59"/><i><font color="Green">E45</font></i>: 
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" target="_self">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:59"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:59"><i><font color="Green">E43</font></i></a>, <a class="ref" href="supinf_1.html#T59" target="_self">Th59</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E8:59"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E9:59"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E6:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="supinf_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E10:59"/><b>then </b>
not  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E11:59"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_1.html#E3:59"><i><font color="Green">E44</font></i></a>, <a class="txt" href="supinf_1.html#E4:59"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T13">XXREAL_0:13</a></i>;<br/>


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