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

<a NAME="E1:34"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> <br/> for <font color="Olive">F</font> being   <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Olive">A</font> holds  <font color="Olive">F</font> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">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">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">A</font>;<br/>



<a NAME="E1:34_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:34_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:34_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="measure5.html#T63">MEASURE5:63</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1"/>
<font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <font color="Olive">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">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> ;<br/>

<a NAME="E1:34_2"/><i><font color="Green">E28</font></i>: 
<a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:34_2"/>
 <a href="supinf_2.html#K1">0.</a>  is    <a href="supinf_1.html#M2">minorant</a> of  <a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">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">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a><br/>

<b>assume </b><a NAME="E1:34_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:34_2_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="measure7.html#K7" target="_self">vol</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:34_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:34_2_1"/><i><font color="Green">E33</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D23">SUPINF_2:def 23</a></i>;<br/>
<b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0;<br/>
<a NAME="E6:34_2_1"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 )
 
<b>by </b><i><a class="txt" href="measure7.html#E4:34"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a>, <a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/>
<a NAME="E7:34_2_1"/><b>then </b>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/>
<a NAME="E8:34_2_1"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#T2" target="_self">Th2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:34_2_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>


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

<b>set </b><font color="Maroon">G</font> = <font color="Maroon">D</font>;<br/>
<a NAME="E4:34"/><i><font color="Green">E35</font></i>: 
 <a href="measure5.html#K6">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E5:34"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> is   <a href="measure5.html#NM1">Interval</a>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:34"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">D</font> as    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of  <a href="measure5.html#K6">{}</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E7:34"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:34"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:34_3"/>
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <a href="measure5.html#K6">{}</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E2:34"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:34_3"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, <a class="ref" href="measure5.html#T60">MEASURE5:60</a></i>;<br/>


</div><b>end;</b></div>
 <a href="measure7.html#K7" target="_self">vol</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0
<b>by </b><i>, <a class="txt" href="measure7.html#E4:34"><i><font color="Green">E35</font></i></a>, <a class="ref" href="supinf_2.html#T67">SUPINF_2:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="txt" href="measure7.html#E4:34"><i><font color="Green">E35</font></i></a></i>
;<br/>
<a NAME="E10:34"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R2">in</a>  <a href="measure7.html#K9" target="_self">Svc</a> <a href="measure5.html#K6">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:34"/>
 <a href="supinf_2.html#K1">0.</a>  is    <a href="supinf_1.html#M2">minorant</a> of  <a href="measure7.html#K9" target="_self">Svc</a> <a href="measure5.html#K6">{}</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">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a><br/>

<b>assume </b><a NAME="E1:34_5"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="measure7.html#K9" target="_self">Svc</a> <a href="measure5.html#K6">{}</a> 
 ;<br/>

<a NAME="E2:34_5"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <a href="measure5.html#K6">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:34_5"/><b>then </b>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <a href="measure5.html#K6">{}</a> </span>)</span> &amp;  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <a href="measure5.html#K6">{}</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="ref" href="supinf_1.html#T79">SUPINF_1:79</a></i>;<br/>
<b>hence </b><a NAME="E4:34_5"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:34"/><b>then </b>
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <a href="measure5.html#K6">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="measure7.html#T2" target="_self">Th2</a>, <a class="ref" href="supinf_1.html#T81">SUPINF_1:81</a></i>;<br/>
<b>hence </b><a NAME="E13:34"/>
<a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <a href="measure5.html#K6">{}</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> , <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> ;<br/>

<b>assume </b><a NAME="E14:34"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E15:34"/><i><font color="Green">E59</font></i>: 
 <a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_6"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E4:34_6"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="measure7.html#K7" target="_self">vol</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:34_6"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="measure7.html#K3" target="_self">.</a> <font color="Olive">n</font> is   <a href="measure5.html#NM1">Interval</a> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E6:34_6"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> as    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">A</font> <b>by </b><i><a class="txt" href="measure7.html#E7:34"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E8:34_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="measure7.html#K7" target="_self">vol</a> <font color="Maroon">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">E68</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> = 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:34_6_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_6_1"/>
<font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:34_6_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E8:34"><i><font color="Green">E42</font></i></a>, <a class="ref" href="supinf_2.html#T62">SUPINF_2:62</a></i>;<br/>
<a NAME="E4:34_6_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> = 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:34_6_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:34_6_1"/>
<font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:34_6_1"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#T4" target="_self">Th4</a>, <a class="ref" href="supinf_2.html#T62">SUPINF_2:62</a></i>;<br/>
<b>thus </b><a NAME="E7:34_6_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="measure7.html#K7" target="_self">vol</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="measure7.html#E10:34"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:34_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E16:34"/>
( <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">A</font></span>)</span> &amp; <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:34"/>
<a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure7.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_1.html#T94">SUPINF_1:94</a></i>;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">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">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span> or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</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">E74</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:34_7_1"/><i><font color="Green">E75</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#D23">SUPINF_2:def 23</a></i>;<br/><a NAME="E3:34_7_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:34_7_1_1"/>
<span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E2:34_7_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:34_7_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/><a NAME="E5:34_7_1"/>
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">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">E78</font></i>: 
not  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">y</font> =  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">x</font> =  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E2:34_7_1_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> 0
 
<b>by </b><i><a class="txt" href="measure7.html#E7:34"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E3:34_7_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E4:34_7_1_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">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#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/>
<a NAME="E6:34_7_1_2"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">eps</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E8:34_7_1_2"/><i><font color="Green">E81</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">eps</font> &amp; <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p4">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">eps</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">eps</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure7.html#E10:34"><i><font color="Green">E43</font></i></a>, <a class="ref" href="measure6.html#T25">MEASURE6:25</a></i>;<br/>
<b>consider </b><font color="Maroon">E</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E10:34_7_1_2"/><i><font color="Green">E82</font></i>: 
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> ) &amp;  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">E</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">eps</font> )
 <b>by </b><i>, <a class="ref" href="measure6.html#T31">MEASURE6:31</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span>] <b>means </b> ex <font color="Olive">F0</font> being   <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">F0</font> &amp;  <a href="measure7.html#K7" target="_self">vol</a> <font color="Olive">F0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E11:34_7_1_2"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">F0</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:34_7_1_2_1"/><i><font color="Green">E84</font></i>: 
( <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:34_7_1_2_1"/>
 <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:34_7_1_2_1"/><b>then </b>
not  <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T39">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" target="_self">Svc</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:34_7_1_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="measure7.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E8:34_7_1_2_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="measure7.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:34_7_1_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:34_7_1_2_1"/><i><font color="Green">E89</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T79">SUPINF_1:79</a></i>;<br/>
<a NAME="E11:34_7_1_2_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/>
<a NAME="E12:34_7_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="measure7.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E13:34_7_1_2_1"/><b>then </b>
(  <a href="supinf_2.html#K1">0.</a>  is   <a href="real_1.html#NM1">Real</a> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp;  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E14:34_7_1_2_1"/><b>then </b>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> &amp;  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i><a class="txt" href="measure7.html#E14:34"><i><font color="Green">E46</font></i></a>, <a class="ref" href="measure6.html#T17">MEASURE6:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">S</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E16:34_7_1_2_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">S</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="measure6.html#T32">MEASURE6:32</a></i>;<br/>
<b>consider </b><font color="Maroon">FS</font> being    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E18:34_7_1_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="measure7.html#K7" target="_self">vol</a> <font color="Maroon">FS</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">FS</font> = <font color="Maroon">FS</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">FS</font>
;<br/>

<b>thus </b><a NAME="E20:34_7_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">FS</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">FF</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span><b> such that </b><br/><a NAME="E13:34_7_1_2"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="measure7.html#T6" target="_self">Th6</a>);<br/></i>
<a NAME="E14:34_7_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> is    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">F0</font> being    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E2:34_7_1_2_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp;  <a href="measure7.html#K7" target="_self">vol</a> <font color="Maroon">F0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:34_7_1_2_2"/>
<font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="measure7.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">FF</font> as    <a href="measure7.html#M2" target="_self">Interval_Covering</a> of <font color="Maroon">F</font> <b>by </b><i/>;<br/>
<a NAME="E16:34_7_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E14:34"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E17:34_7_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">E</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E18:34_7_1_2"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">G0</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E20:34_7_1_2"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E21:34_7_1_2"/><i><font color="Green">E97</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">G0</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <font color="Maroon">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E8:34"><i><font color="Green">E42</font></i></a>, <a class="ref" href="measure7.html#T7" target="_self">Th7</a>, <a class="ref" href="measure7.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E22:34_7_1_2"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">FF</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">F0</font> being    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E2:34_7_1_2_3"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">F0</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">FF</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="measure7.html#K7" target="_self">vol</a> <font color="Maroon">F0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E3:34_7_1_2_3"/>
<span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E4:34_7_1_2_3"/><b>then </b>
 <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:34_7_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">FF</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:34_7_1_2_3"/>
<span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">FF</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="measure7.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:34_7_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">FF</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E24:34_7_1_2"/><b>then </b>
 <a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">FF</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E25:34_7_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">FF</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">G0</font>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T62">SUPINF_2:62</a></i>;<br/>
<a NAME="E26:34_7_1_2"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="txt" href="measure7.html#E8:34"><i><font color="Green">E42</font></i></a>, <a class="ref" href="measure7.html#T7" target="_self">Th7</a>, <a class="ref" href="measure6.html#T2">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#K19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <font color="Maroon">E</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">eps</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E14:34"><i><font color="Green">E46</font></i></a>, <a class="ref" href="measure1.html#T4">MEASURE1:4</a></i>;<br/>
<a NAME="E28:34_7_1_2"/><b>then </b>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">G0</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">eps</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E29:34_7_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">FF</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">eps</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E30:34_7_1_2"/>
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="measure7.html#K9" target="_self">Svc</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">FF</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E31:34_7_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="measure7.html#T8" target="_self">Th8</a>, <a class="ref" href="xxreal_0.html#T2">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" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, </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">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:34_7_2"/><i><font color="Green">E102</font></i>: 
 <a href="measure7.html#K9" target="_self">Svc</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E4:34_7_2"/>
 <a href="supinf_1.html#K7">inf</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T86">SUPINF_1:86</a></i>;<br/><a NAME="E5:34_7_2"/><b>then </b>
<a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E6:34_7_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E7:34_7_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:34_7_2_1"/>
<span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E2:34_7_2_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E8:34_7_2"/><b>then </b>
<a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/><a NAME="E9:34_7_2"/><b>then </b>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T64">SUPINF_2:64</a></i>;<br/><b>hence </b><a NAME="E10:34_7_2"/>
<a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K11" target="_self">OS_Meas</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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