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<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:12"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:12"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E4:12"/>
( <font color="Maroon">g</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E5:12"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="integra1.html#T13">INTEGRA1:13</a>, <a class="ref" href="integra1.html#T15">INTEGRA1:15</a></i>;<br/>
<a NAME="E6:12"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:12"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>


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