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<b>let </b><font color="Maroon" title="c2">F</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="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">G</font> be    <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Maroon" title="c2">F</font>;<br/><b>let </b><font color="Maroon" title="c4">n</font> be    <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> ;<br/>





<a NAME="E1:24"/><i><font color="Green" title="E23">A1</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#D7" target="_self" title="MEASURE7:def.7">Def7</a></i>;<br/>
<a NAME="E2:24"/>
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   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be    <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> ;<br/>

<a NAME="E1:24_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure5.html#T63" title="MEASURE5:th.63">MEASURE5:63</a></i>;<br/>
<b>hence </b><a NAME="E2:24_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:24"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></i>;<br/>
<b>hence </b><a NAME="E4:24"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="measure7.html#E1:24"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="measure6.html#T2" title="MEASURE6:th.2">MEASURE6:2</a></i>;<br/>


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