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<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">G</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">G</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 ;<br/>

<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">H</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">S</font>, <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E2:13"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">H</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>assume </b><a NAME="E3:13"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">k</font> ) &amp; ( not <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 ;<br/>

<a NAME="E4:13"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#R2">=</a>  <a href="measure7.html#K2" target="_self">On</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:13"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">G</font>
 <b>by </b><i>, , </i>;<br/>


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