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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="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</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#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon" title="c3">F</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a>
 ;<br/><a NAME="E2:59_1"/><b>then </b>
 <a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c3">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def11</a></i>;<br/><a NAME="E3:59_1"/><b>then </b><i><font color="Green" title="E46">A1</font></i>: 
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T16" target="_self" title="SUPINF_2:th.16">Th16</a></i>;<br/><a NAME="E4:59_1"/>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#D3" target="_self" title="SUPINF_2:def.3">Def3</a></i>;<br/><a NAME="E5:59_1"/><b>then </b>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="supinf_2.html#E3:59_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="supinf_2.html#T38" target="_self" title="SUPINF_2:th.38">Th38</a></i>;<br/><b>hence </b><a NAME="E6:59_1"/>
 <a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font> is <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#D12" target="_self" title="SUPINF_2:def.12">Def12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:59"/>
 <a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font> is <a href="supinf_2.html#V2" title="SUPINF_2:attr.2">bounded_below</a>
 ;<br/>

<a NAME="E3:59"/><b>then </b>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D12" target="_self" title="SUPINF_2:def.12">Def12</a></i>;<br/>
<a NAME="E4:59"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K11" title="SUPINF_2:func.11">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K14" title="SUPINF_2:func.14">-</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#T16" target="_self" title="SUPINF_2:th.16">Th16</a></i>;<br/>
<a NAME="E5:59"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K10" title="SUPINF_2:func.10">sup</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#T38" target="_self" title="SUPINF_2:th.38">Th38</a></i>;<br/>
<b>hence </b><a NAME="E6:59"/>
<font color="Maroon" title="c3">F</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#D11" target="_self" title="SUPINF_2:def.11">Def11</a>, <a class="ref" href="supinf_2.html#T4" target="_self" title="SUPINF_2:th.4">Th4</a></i>;<br/>


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