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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/>

<a NAME="E1:50"/>
( not <font color="Maroon">c<sub>1</sub></font> is <a href="supinf_1.html#V3" target="_self">bounded_above</a> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>2</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="supinf_1.html#V3" target="_self">bounded_above</a>
 ;<br/>

<b>take </b>
 <a href="supinf_1.html#K1" target="_self">+infty</a> 
;<br/>

<b>thus </b><a NAME="E2:50_1"/>
(  <a href="supinf_1.html#K1" target="_self">+infty</a>  is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="supinf_1.html#K1" target="_self">+infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:50_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="supinf_1.html#T33" target="_self">Th33</a>, <a class="ref" href="supinf_1.html#T70" target="_self">Th70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>2</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="supinf_1.html#T68" target="_self">Th68</a></i>;<br/>


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