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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:46"/>
<font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V3" title="SUPINF_1:attr.3">bounded_above</a>
 ;<br/>

<a NAME="E2:46"/><b>then </b>
( ( <font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V3" title="SUPINF_1:attr.3">bounded_above</a> &amp; not <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) or ( <font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V3" title="SUPINF_1:attr.3">bounded_above</a> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )
 
;<br/>
<b>hence </b><a NAME="E3:46"/>
 ex <font color="Olive" title="b1">UB</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">UB</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b2">y</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> ) )
 <b>by </b><i><a class="ref" href="supinf_1.html#T62" target="_self" title="SUPINF_1:th.62">Th62</a>, <a class="ref" href="supinf_1.html#T66" target="_self" title="SUPINF_1:th.66">Th66</a></i>;<br/>


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