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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:38"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:38"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E3:38"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E4:38"/><i><font color="Green">E55</font></i>: 
 <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>, </i>;<br/>
<a NAME="E5:38"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:38"/>
 <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#D3">SEQ_4:def 3</a></i>;<br/>


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