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

<b>let </b><font color="Maroon">c<sub>3</sub></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/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="measure7.html#M2" target="_self">Interval_Covering</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span><b> such that </b><br/><a NAME="E2:33"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="funcop_1.html#K9">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> )
 <b>by </b><i><a class="ref" href="measure6.html#T1">MEASURE6:1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>
<a NAME="E3:33"/>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <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/>
<a NAME="E4:33"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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"><span class="p4">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:33"/><i><font color="Green">E32</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">c<sub>4</sub></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#K8" target="_self">vol</a> <font color="Maroon">c<sub>4</sub></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="E6:33"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> <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"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <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#K8" target="_self">vol</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon">c<sub>8</sub></font> <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"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="measure7.html#E3:33_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:33_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <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/>for <font color="Olive">b<sub>2</sub></font> being   <a href="measure7.html#M2" target="_self">Interval_Covering</a> of <font color="Olive">b<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <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#K8" target="_self">vol</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="measure7.html#E2:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="measure7.html#T15" target="_self">Th15</a></i>;<br/>
<b>take </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#K8" target="_self">vol</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
;<br/>

<a NAME="E7:33_1"/>
(  <a href="relset_1.html#K4">dom</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#K8" target="_self">vol</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:33_1"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <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#K8" target="_self">vol</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <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#K8" target="_self">vol</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="measure7.html#E3:33_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="measure7.html#E6:33_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:33"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E4:33"><i><font color="Green">E31</font></i></a>, <a class="txt" href="measure7.html#E5:33"><i><font color="Green">E32</font></i></a>, <a class="ref" href="supinf_1.html#T99">SUPINF_1:99</a></i>;<br/>
<a NAME="E8:33"/>
 <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="measure7.html#K9" target="_self">Svc</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">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E9:33"/><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> <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">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T79">SUPINF_1:79</a></i>;<br/>
<b>hence </b><a NAME="E10:33"/>
 <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">c<sub>3</sub></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">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E7:33"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
