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<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c1">F</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c2">G</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">H</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">S</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E2:13"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c4">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<b>assume </b><a NAME="E3:13"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c1">F</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">k</font> ) &amp; ( not <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c1">F</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) )
 ;<br/>

<a NAME="E4:13"/><b>then </b>
<font color="Maroon" title="c1">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="measure7.html#K2" title="MEASURE7:func.2">On</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#D1" target="_self" title="MEASURE7:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:13"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="measure7.html#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="measure7.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="measure7.html#T11" target="_self" title="MEASURE7:th.11">Th11</a></i>;<br/>


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