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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">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:37"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:37"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E4:37"/><i><font color="Green" title="E28">A4</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:37"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="integra1.html#T13" target="_self" title="INTEGRA1:th.13">Th13</a></i>;<br/>
<a NAME="E5:37"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:37"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="integra1.html#T15" target="_self" title="INTEGRA1:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E6:37"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="integra1.html#E4:37"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>


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