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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#K6" title="NUMBERS:func.6">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="E1:57_1"/>
<font color="Maroon" title="c3">F</font> is <a href="supinf_2.html#V3" title="SUPINF_2:attr.3">bounded</a>
 ;<br/><a NAME="E2:57_1"/><b>then </b>
( <font color="Maroon" title="c3">F</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> &amp; <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#D13" target="_self" title="SUPINF_2:def.13">Def13</a></i>;<br/><b>hence </b><a NAME="E3:57_1"/>
(  <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>  &amp;  <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> <font color="Maroon" title="c3">F</font> )
 <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#D12" target="_self" title="SUPINF_2:def.12">Def12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:57"/>
(  <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>  &amp;  <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> <font color="Maroon" title="c3">F</font> )
 ;<br/>

<a NAME="E3:57"/><b>then </b>
( <font color="Maroon" title="c3">F</font> is <a href="supinf_2.html#V1" title="SUPINF_2:attr.1">bounded_above</a> &amp; <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#D11" target="_self" title="SUPINF_2:def.11">Def11</a>, <a class="ref" href="supinf_2.html#D12" target="_self" title="SUPINF_2:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E4:57"/>
<font color="Maroon" title="c3">F</font> is <a href="supinf_2.html#V3" title="SUPINF_2:attr.3">bounded</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#D13" target="_self" title="SUPINF_2:def.13">Def13</a></i>;<br/>


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