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

<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">G</font> be    <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Maroon" title="c3">F</font>;<br/>



<b>consider </b><font color="Maroon" title="c5">H</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> ,<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span><b> such that </b><br/><a NAME="E2:33"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c5">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )
 <b>by </b><i><a class="ref" href="measure6.html#T1" title="MEASURE6:th.1">MEASURE6:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">GG</font> =  <a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font>;<br/>
<b>set </b><font color="Maroon" title="c7">Q</font> =  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span>;<br/>
<a NAME="E3:33"/>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <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_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <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/>
<a NAME="E4:33"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
 <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:33"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</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="E6:33"/>
for <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</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" title="c8">x</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green" title="E33">A4</font></i>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c9">n</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">n</font> as    <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>by </b><i><a class="txt" href="measure7.html#E3:33_1"><i><font color="Green" title="E33">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">m</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="E6:33_1"/><i><font color="Green" title="E34">A5</font></i>: 
for <font color="Olive" title="b1">F</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="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/> for <font color="Olive" title="b2">G</font> being   <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Olive" title="b1">F</font> holds  <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Olive" title="b2">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <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> <font color="Maroon" title="c10">n</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#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Olive" title="b2">G</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="measure7.html#E2:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="measure7.html#T15" target="_self" title="MEASURE7:th.15">Th15</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c11">m</font>
;<br/>

<a NAME="E7:33_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">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" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">x</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#K18" title="SUPINF_2:func.18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c11">m</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E3:33_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="measure7.html#E6:33_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:33"/><b>then </b><i><font color="Green" title="E33">A6</font></i>: 
 <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E4:33"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="measure7.html#E5:33"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="supinf_1.html#T99" title="SUPINF_1:th.99">SUPINF_1:99</a></i>;<br/>
<a NAME="E8:33"/>
 <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span> <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"><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="c3">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></i>;<br/>
<a NAME="E9:33"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1: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="c3">F</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T79" title="SUPINF_1:th.79">SUPINF_1:79</a></i>;<br/>
<b>hence </b><a NAME="E10:33"/>
 <a href="supinf_1.html#K7" title="SUPINF_1: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="c3">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#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c4">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E7:33"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div>
