<?xml version="1.0"?>
<div class="add">

<a NAME="E1:34"/><i><font color="Green" title="E30">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <br/> for <font color="Olive" title="b2">F</font> being   <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Olive" title="b1">A</font> holds  <font color="Olive" title="b2">F</font> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c3">A</font>;<br/>



<a NAME="E1:34_1"/>
for <font color="Olive" title="b1">n</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"><font color="Maroon" title="c4">F</font> <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">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:34_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <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">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E2:34_1_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"><font color="Maroon" title="c4">F</font> <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">n</font>
 <b>by </b><i><a class="ref" href="measure5.html#T63" title="MEASURE5:th.63">MEASURE5:63</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1"/>
<font color="Maroon" title="c4">F</font> <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/>


</div><b>end;</b></div>
<a NAME="E2:34"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</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> <a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/>

<a NAME="E1:34_2"/><i><font color="Green" title="E32">A3</font></i>: 
<a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></i>;<br/>
<a NAME="E2:34_2"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a><br/>

<b>assume </b><a NAME="E1:34_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">F</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E3:34_2_1"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></i>;<br/>
<a NAME="E4:34_2_1"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c5">F</font> <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="txt" href="measure7.html#E1:34"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E5:34_2_1"/><i><font color="Green" title="E35">A6</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D23" title="SUPINF_2:def.23">SUPINF_2:def 23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">y</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0;<br/>
<a NAME="E6:34_2_1"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> &amp;  <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="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 )
 
<b>by </b><i><a class="txt" href="measure7.html#E4:34_2_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a>, <a class="ref" href="supinf_2.html#T59" title="SUPINF_2:th.59">SUPINF_2:59</a></i>;<br/>
<a NAME="E7:34_2_1"/><b>then </b>
(  <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="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T76" title="SUPINF_1:th.76">SUPINF_1:76</a></i>;<br/>
<b>hence </b><a NAME="E8:34_2_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> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="measure7.html#E3:34_2_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="measure7.html#E5:34_2_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34_2"/>
 <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="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="measure7.html#E1:34_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="supinf_1.html#D17" title="SUPINF_1:def.17">SUPINF_1:def 17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
 <a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="measure4.html#D2" title="MEASURE4:def.2">MEASURE4:def 2</a><br/>

<b>set </b><font color="Maroon" title="c3">G</font> = <font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E4:34"/><i><font color="Green" title="E32">A7</font></i>: 
 <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E5:34"/>
for <font color="Olive" title="b1">n</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  <font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c2">D</font> as    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  <b>by </b><i><a class="txt" href="measure7.html#E4:34"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></i>;<br/>
<a NAME="E7:34"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c4">G</font> <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="txt" href="measure7.html#E1:34"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E8:34"/><i><font color="Green" title="E34">A9</font></i>: 
for <font color="Olive" title="b1">r</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>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <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">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:34_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E2:34_3"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <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">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c4">G</font> = 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0
<b>by </b><i><a class="txt" href="measure7.html#E7:34"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="supinf_2.html#T67" title="SUPINF_2:th.67">SUPINF_2:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0
<b>by </b><i><a class="ref" href="supinf_2.html#T63" title="SUPINF_2:th.63">SUPINF_2:63</a></i>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<b>by </b><i><a class="txt" href="measure7.html#E8:34"><i><font color="Green" title="E34">A9</font></i></a></i>
;<br/>
<a NAME="E10:34"/><b>then </b><i><font color="Green" title="E35">A10</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 
<b>by </b><i><a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></i>;<br/>
<a NAME="E11:34"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a><br/>

<b>assume </b><a NAME="E1:34_5"/><i><font color="Green" title="E36">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/>

<a NAME="E2:34_5"/>
 <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="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E2:34"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<a NAME="E3:34_5"/><b>then </b>
(  <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="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> </span>)</span> &amp;  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> )
 
<b>by </b><i><a class="txt" href="measure7.html#E1:34_5"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a>, <a class="ref" href="supinf_1.html#T79" title="SUPINF_1:th.79">SUPINF_1:79</a></i>;<br/>
<b>hence </b><a NAME="E4:34_5"/>
 <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> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:34"/><b>then </b>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E10:34"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="supinf_1.html#T81" title="SUPINF_1:th.81">SUPINF_1:81</a></i>;<br/>
<b>hence </b><a NAME="E13:34"/>
<a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/>

<b>assume </b><a NAME="E14:34"/><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 ;<br/>

<a NAME="E15:34"/><i><font color="Green" title="E37">A13</font></i>: 
 <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <font color="Maroon" title="c5">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_6"/><i><font color="Green" title="E38">A14</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <font color="Maroon" title="c6">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<b>consider </b><font color="Maroon" title="c9">F</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c6">B</font><b> such that </b><br/><a NAME="E4:34_6"/><i><font color="Green" title="E39">A15</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c9">F</font>
 <b>by </b><i><a class="txt" href="measure7.html#E1:34_6"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></i>;<br/>
<a NAME="E5:34_6"/><i><font color="Green" title="E40">A16</font></i>: 
( <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> &amp; ( for <font color="Olive" title="b1">n</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  <font color="Maroon" title="c9">F</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Olive" title="b1">n</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a> ) )
 
<b>by </b><i><a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></i>;<br/>
<a NAME="E6:34_6"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E14:34"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c9">F</font> as    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c5">A</font> <b>by </b><i><a class="txt" href="measure7.html#E5:34_6"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></i>;<br/>
<a NAME="E8:34_6"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c10">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_6_1"/><i><font color="Green" title="E41">A17</font></i>: 
for <font color="Olive" title="b1">n</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  <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <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">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <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">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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/>

<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <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="c11">n</font> = 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <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="c11">n</font>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
;<br/>
<b>hence </b><a NAME="E2:34_6_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <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="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <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="c11">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_6_1"/>
<font color="Maroon" title="c10">G</font> <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="txt" href="measure7.html#E1:34"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E3:34_6_1"/><b>then </b><i><font color="Green" title="E42">A18</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E1:34_6_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></i>;<br/>
<a NAME="E4:34_6_1"/><i><font color="Green" title="E43">A19</font></i>: 
for <font color="Olive" title="b1">n</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  <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <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">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <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">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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/>

<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <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="c11">n</font> = 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <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="c11">n</font>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
;<br/>
<b>hence </b><a NAME="E2:34_6_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <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="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <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="c11">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:34_6_1"/>
<font color="Maroon" title="c9">F</font> <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="txt" href="measure7.html#E1:34"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E6:34_6_1"/><b>then </b>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E4:34_6_1"><i><font color="Green" title="E43">A19</font></i></a>, <a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></i>;<br/>
<b>hence </b><a NAME="E7:34_6_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="txt" href="measure7.html#E4:34_6"><i><font color="Green" title="E39">A15</font></i></a>, <a class="txt" href="measure7.html#E3:34_6_1"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:34_6"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:34"/>
( <a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp; <a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E17:34"/>
<a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="measure7.html#E15:34"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="supinf_1.html#T94" title="SUPINF_1:th.94">SUPINF_1:94</a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_7"/>
( for <font color="Olive" title="b1">n</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="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  ex <font color="Olive" title="b1">n</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>  st  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1"><b>suppose </b></a><a NAME="E1:34_7_1"/><i><font color="Green" title="E38">A20</font></i>: 
for <font color="Olive" title="b1">n</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="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:34_7_1"/><i><font color="Green" title="E39">A21</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#D23" title="SUPINF_2:def.23">SUPINF_2:def 23</a></i>;<br/><a NAME="E3:34_7_1"/><i><font color="Green" title="E40">A22</font></i>: 
for <font color="Olive" title="b1">n</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"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:34_7_1_1"/>
<span class="p1">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E2:34_7_1_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"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="measure7.html#E2:34"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:34_7_1"/><b>then </b><i><font color="Green" title="E41">A23</font></i>: 
<a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font> 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/><a NAME="E5:34_7_1"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_7_1_2"/><i><font color="Green" title="E42">A24</font></i>: 
not  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">y</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">x</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E2:34_7_1_2"/>
 <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#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0
 
<b>by </b><i><a class="txt" href="measure7.html#E3:34_7_1"><i><font color="Green" title="E40">A22</font></i></a></i>;<br/>
<a NAME="E3:34_7_1_2"/><b>then </b><i><font color="Green" title="E43">A25</font></i>: 
 <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="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63" title="SUPINF_2:th.63">SUPINF_2:63</a></i>;<br/>
<a NAME="E4:34_7_1_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E5:34_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T76" title="SUPINF_1:th.76">SUPINF_1:76</a></i>;<br/>
<a NAME="E6:34_7_1_2"/><b>then </b>
 <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="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E3:34_7_1_2"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">eps</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E8:34_7_1_2"/><i><font color="Green" title="E44">A26</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">eps</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p4">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c10">eps</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">eps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure7.html#E1:34_7_1_2"><i><font color="Green" title="E42">A24</font></i></a>, <a class="ref" href="measure6.html#T25" title="MEASURE6:th.25">MEASURE6:25</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">E</font> being   <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#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E10:34_7_1_2"/><i><font color="Green" title="E45">A27</font></i>: 
( ( for <font color="Olive" title="b1">n</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font> ) &amp;  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c11">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">eps</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E8:34_7_1_2"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="measure6.html#T31" title="MEASURE6:th.31">MEASURE6:31</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><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> </span>)</span>] <b>means </b> ex <font color="Olive" title="b1">F0</font> being   <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> $1 st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F0</font> &amp;  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Olive" title="b1">F0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> $1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> $1</span>)</span> );<br/>
<a NAME="E11:34_7_1_2"/><i><font color="Green" title="E46">A28</font></i>: 
for <font color="Olive" title="b1">n</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>   ex <font color="Olive" title="b2">F0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><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> </span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">F0</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="E1:34_7_1_2_1"/><i><font color="Green" title="E47">A29</font></i>: 
( <a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> &amp;  <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="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="measure7.html#E2:34"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></i>;<br/>
<a NAME="E2:34_7_1_2_1"/>
 <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="measure7.html#E1:34_7_1"><i><font color="Green" title="E38">A20</font></i></a></i>;<br/>
<a NAME="E3:34_7_1_2_1"/><b>then </b>
not  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/>
<a NAME="E4:34_7_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:34_7_1_2_1"/><i><font color="Green" title="E48">A30</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="measure7.html#E6:34_7_1_2_1"><i><font color="Green" title="E48">A30</font></i></a></i>;<br/>
<a NAME="E8:34_7_1_2_1"/><i><font color="Green" title="E49">A31</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> &amp; not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="measure7.html#E6:34_7_1_2_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:34_7_1_2_1"/><b>then </b><i><font color="Green" title="E50">A32</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:34_7_1_2_1"/><i><font color="Green" title="E51">A33</font></i>: 
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E8:34_7_1_2_1"><i><font color="Green" title="E49">A31</font></i></a>, <a class="ref" href="supinf_1.html#T79" title="SUPINF_1:th.79">SUPINF_1:79</a></i>;<br/>
<a NAME="E11:34_7_1_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E9:34_7_1_2_1"><i><font color="Green" title="E50">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<a NAME="E12:34_7_1_2_1"/><b>then </b>
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp;  <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="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> &amp;  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="measure7.html#E1:34_7_1_2_1"><i><font color="Green" title="E47">A29</font></i></a>, <a class="txt" href="measure7.html#E10:34_7_1_2_1"><i><font color="Green" title="E51">A33</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E13:34_7_1_2_1"/><b>then </b>
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c12">n</font> &amp;  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 
<b>by </b><i><a class="txt" href="measure7.html#E10:34_7_1_2"><i><font color="Green" title="E45">A27</font></i></a>, <a class="ref" href="measure6.html#T17" title="MEASURE6:th.17">MEASURE6:17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">S</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E15:34_7_1_2_1"/><i><font color="Green" title="E52">A34</font></i>: 
( <font color="Maroon" title="c15">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> &amp; <font color="Maroon" title="c15">S</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="measure6.html#T32" title="MEASURE6:th.32">MEASURE6:32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">FS</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font><b> such that </b><br/><a NAME="E17:34_7_1_2_1"/><i><font color="Green" title="E53">A35</font></i>: 
<font color="Maroon" title="c15">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c16">FS</font>
 <b>by </b><i><a class="txt" href="measure7.html#E15:34_7_1_2_1"><i><font color="Green" title="E52">A34</font></i></a>, <a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">FS</font> = <font color="Maroon" title="c16">FS</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><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> </span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c17">FS</font>
;<br/>

<b>thus </b><a NAME="E19:34_7_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c17">FS</font>]
 <b>by </b><i><a class="txt" href="measure7.html#E15:34_7_1_2_1"><i><font color="Green" title="E52">A34</font></i></a>, <a class="txt" href="measure7.html#E17:34_7_1_2_1"><i><font color="Green" title="E53">A35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c12">FF</font> being   <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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><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> </span>)</span><b> such that </b><br/><a NAME="E13:34_7_1_2"/><i><font color="Green" title="E47">A36</font></i>: 
for <font color="Olive" title="b1">n</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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c12">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="measure7.html#E11:34_7_1_2"><i><font color="Green" title="E46">A28</font></i></a>);<br/></i>
<a NAME="E14:34_7_1_2"/>
for <font color="Olive" title="b1">n</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  <font color="Maroon" title="c12">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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/>

<b>consider </b><font color="Maroon" title="c14">F0</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c13">n</font><b> such that </b><br/><a NAME="E2:34_7_1_2_2"/><i><font color="Green" title="E48">A37</font></i>: 
( <font color="Maroon" title="c14">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> &amp;  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c14">F0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="measure7.html#E13:34_7_1_2"><i><font color="Green" title="E47">A36</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:34_7_1_2_2"/>
<font color="Maroon" title="c12">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 <b>by </b><i><a class="txt" href="measure7.html#E2:34_7_1_2_2"><i><font color="Green" title="E48">A37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">FF</font> = <font color="Maroon" title="c12">FF</font> as    <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <b>by </b><i><a class="ref" href="measure7.html#D3" target="_self" title="MEASURE7:def.3">Def3</a></i>;<br/>
<a NAME="E16:34_7_1_2"/>
for <font color="Olive" title="b1">n</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> <font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E10:34_7_1_2"><i><font color="Green" title="E45">A27</font></i></a></i>;<br/>
<a NAME="E17:34_7_1_2"/><b>then </b><i><font color="Green" title="E48">A38</font></i>: 
<font color="Maroon" title="c11">E</font> 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>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> $1</span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> $1</span>)</span>;<br/>
<a NAME="E18:34_7_1_2"/><i><font color="Green" title="E49">A39</font></i>: 
for <font color="Olive" title="b1">n</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>   ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c14">G0</font> being   <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#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E20:34_7_1_2"/><i><font color="Green" title="E50">A40</font></i>: 
for <font color="Olive" title="b1">n</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  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c14">G0</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="measure7.html#E18:34_7_1_2"><i><font color="Green" title="E49">A39</font></i></a>);<br/></i>
<a NAME="E21:34_7_1_2"/><i><font color="Green" title="E51">A41</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c14">G0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c11">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E4:34_7_1"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="measure7.html#E17:34_7_1_2"><i><font color="Green" title="E48">A38</font></i></a>, <a class="txt" href="measure7.html#E20:34_7_1_2"><i><font color="Green" title="E50">A40</font></i></a>, <a class="ref" href="measure7.html#T4" target="_self" title="MEASURE7:th.4">Th4</a></i>;<br/>
<a NAME="E22:34_7_1_2"/><i><font color="Green" title="E52">A42</font></i>: 
for <font color="Olive" title="b1">n</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  <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c13">FF</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">G0</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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/>

<b>consider </b><font color="Maroon" title="c16">F0</font> being    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c15">n</font><b> such that </b><br/><a NAME="E2:34_7_1_2_3"/><i><font color="Green" title="E53">A43</font></i>: 
( <font color="Maroon" title="c16">F0</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">FF</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c15">n</font> &amp;  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <font color="Maroon" title="c16">F0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="measure7.html#E13:34_7_1_2"><i><font color="Green" title="E47">A36</font></i></a></i>;<br/>
<a NAME="E3:34_7_1_2_3"/>
<span class="p1">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E4:34_7_1_2_3"/><b>then </b>
 <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FF</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:34_7_1_2_3"><i><font color="Green" title="E53">A43</font></i></a>, <a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></i>;<br/>
<a NAME="E5:34_7_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c13">FF</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">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/>
<b>hence </b><a NAME="E6:34_7_1_2_3"/>
<span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c13">FF</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">G0</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">n</font>
 <b>by </b><i><a class="txt" href="measure7.html#E20:34_7_1_2"><i><font color="Green" title="E50">A40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:34_7_1_2"/>
for <font color="Olive" title="b1">n</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"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c13">FF</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T14" target="_self" title="MEASURE7:th.14">Th14</a></i>;<br/>
<a NAME="E24:34_7_1_2"/><b>then </b>
 <a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c13">FF</font> 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/>
<a NAME="E25:34_7_1_2"/><b>then </b><i><font color="Green" title="E53">A44</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c13">FF</font></span>)</span> <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="c14">G0</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E22:34_7_1_2"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></i>;<br/>
<a NAME="E26:34_7_1_2"/>
(  <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="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> &amp;  <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="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c11">E</font> )
 
<b>by </b><i><a class="txt" href="measure7.html#E4:34_7_1"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="measure7.html#E17:34_7_1_2"><i><font color="Green" title="E48">A38</font></i></a>, <a class="ref" href="measure6.html#T2" title="MEASURE6:th.2">MEASURE6:2</a></i>;<br/>
<a NAME="E27:34_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c11">E</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c10">eps</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E10:34_7_1_2"><i><font color="Green" title="E45">A27</font></i></a>, <a class="ref" href="measure1.html#T4" title="MEASURE1:th.4">MEASURE1:4</a></i>;<br/>
<a NAME="E28:34_7_1_2"/><b>then </b>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <font color="Maroon" title="c14">G0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c10">eps</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E21:34_7_1_2"><i><font color="Green" title="E51">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E29:34_7_1_2"/><b>then </b><i><font color="Green" title="E54">A45</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c13">FF</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c10">eps</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E25:34_7_1_2"><i><font color="Green" title="E53">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E30:34_7_1_2"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c13">FF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#T16" target="_self" title="MEASURE7:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E31:34_7_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure7.html#E2:34_7_1"><i><font color="Green" title="E39">A21</font></i></a>, <a class="txt" href="measure7.html#E8:34_7_1_2"><i><font color="Green" title="E44">A26</font></i></a>, <a class="txt" href="measure7.html#E29:34_7_1_2"><i><font color="Green" title="E54">A45</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:34_7_1"/>
<a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E2:34_7_1"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_2"><b>suppose </b></a><a NAME="E1:34_7_2"/>
 ex <font color="Olive" title="b1">n</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>  st  <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">n</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> <b> such that </b><br/><a NAME="E3:34_7_2"/><i><font color="Green" title="E38">A46</font></i>: 
 <a href="measure7.html#K9" title="MEASURE7:func.9">Svc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><a NAME="E4:34_7_2"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T86" title="SUPINF_1:th.86">SUPINF_1:86</a></i>;<br/><a NAME="E5:34_7_2"/><b>then </b>
<a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure7.html#E3:34_7_2"><i><font color="Green" title="E38">A46</font></i></a>, <a class="ref" href="measure7.html#D10" target="_self" title="MEASURE7:def.10">Def10</a></i>;<br/><a NAME="E6:34_7_2"/><b>then </b><i><font color="Green" title="E39">A47</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/><a NAME="E7:34_7_2"/>
for <font color="Olive" title="b1">n</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"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="E1:34_7_2_1"/>
<span class="p1">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E2:34_7_2_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"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="measure7.html#E2:34"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:34_7_2"/><b>then </b>
<a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font> 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/><a NAME="E9:34_7_2"/><b>then </b>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure7.html#E6:34_7_2"><i><font color="Green" title="E39">A47</font></i></a>, <a class="ref" href="supinf_2.html#T64" title="SUPINF_2:th.64">SUPINF_2:64</a></i>;<br/><b>hence </b><a NAME="E10:34_7_2"/>
<a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="measure7.html#K11" title="MEASURE7:func.11">OS_Meas</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
