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<a NAME="E1:27"/>
for <font color="Olive">b<sub>1</sub></font> being 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>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  holds <br/> not <font color="Olive">b<sub>1</sub></font> is <a href="supinf_1.html#V4" target="_self">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<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="E1:27_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 ;<br/>

<b>assume </b><a NAME="E2:27_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="supinf_1.html#V4" target="_self">bounded_below</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green">E19</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#D12" target="_self">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="supinf_1.html#E4:27_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></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="E7:27_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T4">XREAL_1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</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/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E11:27_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:27_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a>;<br/>
<b>thus </b><a NAME="E13:27_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_1.html#E7:27_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="supinf_1.html#E11:27_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27"/>
not  <a href="supinf_1.html#K3" target="_self">REAL</a>  is <a href="supinf_1.html#V4" target="_self">bounded_below</a>
 ;<br/>


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