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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <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>   st <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  holds <br/> <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> </span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  implies  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> </span><br/>

<a NAME="E2:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T15" target="_self" title="INTEGRA1:th.15">Th15</a></span>;<br/>
<a NAME="E3:36"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="integra1.html#T13" target="_self" title="INTEGRA1:th.13">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:36"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E25">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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