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



<b>assume </b><a NAME="E1:9"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">G</font> is <a href="supinf_2.html#V6">nonnegative</a>
 ;<br/>

<a NAME="E2:9"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_2"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
 <a href="measure7.html#K2" target="_self">On</a> <font color="Maroon">G</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>


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