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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">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<a NAME="E1:5"/>
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E3:5"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:5"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T15">SEQ_4:15</a></i>;<br/>
<a NAME="E5:5"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E6:5"/><b>then </b>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="integra1.html#T14">INTEGRA1:14</a>, <a class="ref" href="integra1.html#T16">INTEGRA1:16</a></i>;<br/>
<a NAME="E7:5"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_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/>
<b>hence </b><a NAME="E8:5"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span> &amp; <font color="Olive">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>


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