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

<b>let </b><font color="Maroon" title="c3">H</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> ,<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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c3">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <font color="Maroon" title="c3">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>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c2">D</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> ex <font color="Olive" title="b1">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>  st <br/>for <font color="Olive" title="b2">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="b3">G</font> being   <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Olive" title="b2">F</font> holds  <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Olive" title="b3">G</font>,<font color="Maroon" title="c3">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> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Olive" title="b3">G</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">m</font>;<br/>
<a NAME="E4:32"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c5">m</font> = <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0;<br/>

<b>let </b><font color="Maroon" title="c6">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="c7">G</font> be    <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Maroon" title="c6">F</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c8">GG</font> =  <a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c3">H</font> as    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> ;<br/>
<a NAME="E2:32_1"/><i><font color="Green" title="E31">A4</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">GG</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GG</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63" title="SUPINF_2:th.63">SUPINF_2:63</a></i>;<br/>
<a NAME="E3:32_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">GG</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">GG</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>;<br/>
<a NAME="E4:32_1"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">GG</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="measure7.html#D13" target="_self" title="MEASURE7:def.13">Def13</a></i>;<br/>
<a NAME="E5:32_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure7.html#T13" target="_self" title="MEASURE7:th.13">Th13</a></i>;<br/>
<a NAME="E6:32_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">GG</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <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"><font color="Maroon" title="c7">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E4:32_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="measure6.html#T3" title="MEASURE6:th.3">MEASURE6:3</a></i>;<br/>
<a NAME="E7:32_1"/><b>then </b><i><font color="Green" title="E33">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">GG</font> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#D7" target="_self" title="MEASURE7:def.7">Def7</a></i>;<br/>
<a NAME="E8:32_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T14" target="_self" title="MEASURE7:th.14">Th14</a></i>;<br/>
<a NAME="E9:32_1"/><b>then </b>
 <a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c7">G</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></i>;<br/>
<a NAME="E10:32_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">m</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#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T2" target="_self" title="MEASURE7:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E11:32_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c3">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> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="measure7.html#E2:32_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="measure7.html#E7:32_1"><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><b>end;</b></div>
<a NAME="E5:32"/><i><font color="Green" title="E31">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>given </b><font color="Maroon" title="c6">m0</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><a NAME="E2:32_2"/><i><font color="Green" title="E32">A8</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#K20" title="SUPINF_2:func.20">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="c3">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="c5">k</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#K20" title="SUPINF_2:func.20">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="c6">m0</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">m0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>

<b>let </b><font color="Maroon" title="c8">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="c9">G</font> be    <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Maroon" title="c8">F</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span>] <b>means </b>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies for <font color="Olive" title="b1">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>  holds  $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> $1</span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Olive" title="b1">m</font> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies for <font color="Olive" title="b1">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>  holds  $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  ) );<br/>
<a NAME="E3:32_2"/><i><font color="Green" title="E33">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_1_1"/>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_1_1"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c10">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">y</font>
;<br/><b>thus </b><a NAME="E3:32_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c11">y</font>]
 <b>by </b><i><a class="txt" href="measure7.html#E1:32_2_1_1_1"><i><font color="Green" title="E34">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_1_2"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c4">y</font>
;<br/><b>thus </b><a NAME="E2:32_2_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c4">y</font>]
 <b>by </b><i><a class="txt" href="measure7.html#E1:32_2_1_1_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">G0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span><b> such that </b><br/><a NAME="E5:32_2"/><i><font color="Green" title="E34">A12</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c10">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="measure7.html#E3:32_2"><i><font color="Green" title="E33">A9</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span>] <b>means </b>( ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies for <font color="Olive" title="b1">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>  holds  $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> $1</span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Olive" title="b1">m</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies for <font color="Olive" title="b1">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>  holds  $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  ) );<br/>
<a NAME="E6:32_2"/><i><font color="Green" title="E35">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_2_1"/>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_2_1_1"><b>suppose </b></a><a NAME="E1:32_2_2_1_1"/><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c11">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">y</font>
;<br/><b>thus </b><a NAME="E3:32_2_2_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">y</font>]
 <b>by </b><i><a class="txt" href="measure7.html#E1:32_2_2_1_1"><i><font color="Green" title="E36">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_2_1_2"><b>suppose </b></a><a NAME="E1:32_2_2_1_2"/><i><font color="Green" title="E36">A15</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c4">y</font>
;<br/><b>thus </b><a NAME="E2:32_2_2_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c4">y</font>]
 <b>by </b><i><a class="txt" href="measure7.html#E1:32_2_2_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c11">G1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span><b> such that </b><br/><a NAME="E8:32_2"/><i><font color="Green" title="E36">A16</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c11">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="measure7.html#E6:32_2"><i><font color="Green" title="E35">A13</font></i></a>);<br/></i>
<a NAME="E9:32_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon" title="c13">f0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:32_2_3"/><i><font color="Green" title="E37">A17</font></i>: 
( <font color="Maroon" title="c10">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f0</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">f0</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">f0</font> = <font color="Maroon" title="c13">f0</font> as   <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>  <b>by </b><i><a class="txt" href="measure7.html#E2:32_2_3"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E4:32_2_3"/>
<font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E5:32_2_3"/><b>then </b><i><font color="Green" title="E38">A18</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">f0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E6:32_2_3"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c14">f0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">s</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_3_1_1"/>
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:32_2_3_1_1_1"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:32_2_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">f0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c15">s</font>
 <b>by </b><i><a class="txt" href="measure7.html#E5:32_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="measure7.html#E2:32_2_3"><i><font color="Green" title="E37">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E3:32_2_3_1_1_1"/>
<font color="Maroon" title="c14">f0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c15">s</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:32_2_3_1_1_2"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_3_1_1_2"/>
<font color="Maroon" title="c14">f0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c15">s</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <b>by </b><i><a class="txt" href="measure7.html#E5:32_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="measure7.html#E2:32_2_3"><i><font color="Green" title="E37">A17</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c15">f0</font> = <font color="Maroon" title="c14">f0</font> as    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <b>by </b><i><a class="txt" href="measure7.html#E5:32_2_3"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></i>;<br/>
<a NAME="E8:32_2_3"/>
<font color="Maroon" title="c10">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f0</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32_2_3"><i><font color="Green" title="E37">A17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:32_2_3"/>
<font color="Maroon" title="c10">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">G0</font> = <font color="Maroon" title="c10">G0</font> as    <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="ref" href="measure7.html#D3" target="_self" title="MEASURE7:def.3">Def3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">GG0</font> =  <a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">H</font>;<br/>
<a NAME="E11:32_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon" title="c15">f0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:32_2_4"/><i><font color="Green" title="E37">A19</font></i>: 
( <font color="Maroon" title="c11">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f0</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">f0</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">f0</font> = <font color="Maroon" title="c15">f0</font> as   <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>  <b>by </b><i><a class="txt" href="measure7.html#E2:32_2_4"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E4:32_2_4"/>
<font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E5:32_2_4"/><b>then </b><i><font color="Green" title="E38">A20</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">f0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E6:32_2_4"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c16">f0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">s</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_4_1_1"/>
( <font color="Maroon" title="c14">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:32_2_4_1_1_1"/>
<font color="Maroon" title="c14">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:32_2_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">f0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c17">s</font>
 <b>by </b><i><a class="txt" href="measure7.html#E8:32_2"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="measure7.html#E2:32_2_4"><i><font color="Green" title="E37">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E3:32_2_4_1_1_1"/>
<font color="Maroon" title="c16">f0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">s</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:32_2_4_1_1_2"/>
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_4_1_1_2"/>
<font color="Maroon" title="c16">f0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">s</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <b>by </b><i><a class="txt" href="measure7.html#E8:32_2"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="measure7.html#E2:32_2_4"><i><font color="Green" title="E37">A19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c17">f0</font> = <font color="Maroon" title="c16">f0</font> as    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <b>by </b><i><a class="txt" href="measure7.html#E5:32_2_4"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></i>;<br/>
<a NAME="E8:32_2_4"/>
<font color="Maroon" title="c11">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f0</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32_2_4"><i><font color="Green" title="E37">A19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:32_2_4"/>
<font color="Maroon" title="c11">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font> is    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c2">D</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">G1</font> = <font color="Maroon" title="c11">G1</font> as    <a href="measure7.html#M2" title="MEASURE7:mode.2">Interval_Covering</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="ref" href="measure7.html#D3" target="_self" title="MEASURE7:def.3">Def3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">GG1</font> =  <a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">H</font>;<br/>
<a NAME="E13:32_2"/><i><font color="Green" title="E37">A21</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:32_2_5"/><i><font color="Green" title="E38">A22</font></i>: 
( <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="measure7.html#D7" target="_self" title="MEASURE7:def.7">Def7</a></i>;<br/>
<a NAME="E2:32_2_5"/>
 <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_2_5_1"/><i><font color="Green" title="E39">A23</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="measure7.html#T13" target="_self" title="MEASURE7:th.13">Th13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_5_1_1"/>
( <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c16">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:32_2_5_1_1_1"/><i><font color="Green" title="E40">A24</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:32_2_5_1_1_1"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</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> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_2_5_1_1_1_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:32_2_5_1_1_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E8:32_2"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_5_1_1_1"><i><font color="Green" title="E40">A24</font></i></a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a></i>;<br/>
<b>hence </b><a NAME="E2:32_2_5_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_2_5_1_1_1_1"/><b>then </b><i><font color="Green" title="E41">A25</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</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_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="ref" href="measure7.html#T1" target="_self" title="MEASURE7:th.1">Th1</a></i>;<br/>
<a NAME="E3:32_2_5_1_1_1_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font> = 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c17">s</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
<br/>.= 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c17">s</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E5:32_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_5_1_1_1"><i><font color="Green" title="E40">A24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
;<br/>
<b>hence </b><a NAME="E2:32_2_5_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_2_5_1_1_1_1"/><b>then </b><i><font color="Green" title="E42">A26</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E1:32_2_5_1"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></i>;<br/>
<a NAME="E5:32_2_5_1_1_1_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font> = 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c17">s</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
<br/>.= 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c17">s</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E5:32_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_5_1_1_1"><i><font color="Green" title="E40">A24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
;<br/>
<b>hence </b><a NAME="E2:32_2_5_1_1_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:32_2_5_1_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E1:32_2_5_1"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></i>;<br/>
<a NAME="E7:32_2_5_1_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</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_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E4:32_2_5_1_1_1_1"><i><font color="Green" title="E42">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E8:32_2_5_1_1_1_1"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</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> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E2:32_2_5_1_1_1_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:32_2_5_1_1_1"/>
 <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:32_2_5_1_1_2"/><i><font color="Green" title="E40">A27</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:32_2_5_1_1_2"/><i><font color="Green" title="E41">A28</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:32_2_5_1_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c17">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c17">s</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E2:32_2_5_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font>
 <b>by </b><i><a class="txt" href="measure7.html#E8:32_2"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_5_1_1_2"><i><font color="Green" title="E40">A27</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_2_5_1_1_2"/><i><font color="Green" title="E42">A29</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:32_2_5_1_1_2_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E5:32_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_5_1_1_2"><i><font color="Green" title="E40">A27</font></i></a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a></i>;<br/>
<b>hence </b><a NAME="E2:32_2_5_1_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:32_2_5_1_1_2"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</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> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_2_5_1_1_2_3"/><i><font color="Green" title="E43">A30</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="measure7.html#T13" target="_self" title="MEASURE7:th.13">Th13</a></i>;<br/>
<a NAME="E2:32_2_5_1_1_2_3"/><i><font color="Green" title="E44">A31</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</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_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E3:32_2_5_1_1_2"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="measure7.html#T1" target="_self" title="MEASURE7:th.1">Th1</a></i>;<br/>
<a NAME="E3:32_2_5_1_1_2_3"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32_2_5_1_1_2"><i><font color="Green" title="E41">A28</font></i></a></i>;<br/>
<a NAME="E4:32_2_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E45">A32</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E1:32_2_5_1_1_2_3"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></i>;<br/>
<a NAME="E5:32_2_5_1_1_2_3"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32_2_5_1_1_2"><i><font color="Green" title="E41">A28</font></i></a></i>;<br/>
<a NAME="E6:32_2_5_1_1_2_3"/><b>then </b>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E1:32_2_5_1"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></i>;<br/>
<a NAME="E7:32_2_5_1_1_2_3"/><b>then </b>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</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_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E4:32_2_5_1_1_2_3"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E8:32_2_5_1_1_2_3"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</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> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E2:32_2_5_1_1_2_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:32_2_5_1_1_2"/>
 <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <font color="Maroon" title="c16">n</font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_2_5"/>
<span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E1:32_2_5"><i><font color="Green" title="E38">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:32_2"/><i><font color="Green" title="E38">A33</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure7.html#T13" target="_self" title="MEASURE7:th.13">Th13</a></i>;<br/>
<a NAME="E15:32_2"/><i><font color="Green" title="E39">A34</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure7.html#T13" target="_self" title="MEASURE7:th.13">Th13</a></i>;<br/>
<a NAME="E16:32_2"/><i><font color="Green" title="E40">A35</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E14:32_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="measure7.html#T7" target="_self" title="MEASURE7:th.7">Th7</a></i>;<br/>
<a NAME="E17:32_2"/><i><font color="Green" title="E41">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure7.html#T13" target="_self" title="MEASURE7:th.13">Th13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">N0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> </span>}</span> ;<br/>
<a NAME="E18:32_2"/><i><font color="Green" title="E42">A37</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">s1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_2_6"/>
<font color="Maroon" title="c17">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> </span>}</span> 
 ;<br/>

<a NAME="E2:32_2_6"/><b>then </b>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s1</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> )
 
;<br/>
<b>hence </b><a NAME="E3:32_2_6"/>
<font color="Maroon" title="c17">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:32_2"/>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">N0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <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>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="measure7.html#E18:32_2"><i><font color="Green" title="E42">A37</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c17">N0</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<a NAME="E21:32_2"/><i><font color="Green" title="E43">A38</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c17">N0</font>  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c18">SOS</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c17">N0</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E23:32_2"/><i><font color="Green" title="E44">A39</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c17">N0</font> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c18">SOS</font> <a href="measure7.html#K1" title="MEASURE7:func.1">.</a> <font color="Olive" title="b1">s</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="measure7.html#E21:32_2"><i><font color="Green" title="E43">A38</font></i></a>);<br/></i>
<a NAME="E24:32_2"/>
for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N0</font> &amp; <font color="Olive" title="b2">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N0</font> &amp; <font color="Maroon" title="c18">SOS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">SOS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s2</font> holds <br/><font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">s1</font>, <font color="Maroon" title="c20">s2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_2_7"/><i><font color="Green" title="E45">A40</font></i>: 
( <font color="Maroon" title="c19">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N0</font> &amp; <font color="Maroon" title="c20">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N0</font> &amp; <font color="Maroon" title="c18">SOS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">SOS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c21">s1</font> = <font color="Maroon" title="c19">s1</font>, <font color="Maroon" title="c22">s2</font> = <font color="Maroon" title="c20">s2</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>  ;<br/>
<b>consider </b><font color="Maroon" title="c23">s11</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="E4:32_2_7"/><i><font color="Green" title="E46">A41</font></i>: 
( <font color="Maroon" title="c23">s11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">s1</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">s11</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E1:32_2_7"><i><font color="Green" title="E45">A40</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">s22</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="E6:32_2_7"/><i><font color="Green" title="E47">A42</font></i>: 
( <font color="Maroon" title="c24">s22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s2</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">s22</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E1:32_2_7"><i><font color="Green" title="E45">A40</font></i></a></i>;<br/>
<a NAME="E7:32_2_7"/><i><font color="Green" title="E48">A43</font></i>: 
( <font color="Maroon" title="c18">SOS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s1</font> &amp; <font color="Maroon" title="c18">SOS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s2</font> )
 
<b>by </b><i><a class="txt" href="measure7.html#E23:32_2"><i><font color="Green" title="E44">A39</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_7"><i><font color="Green" title="E45">A40</font></i></a></i>;<br/>
<a NAME="E8:32_2_7"/>
( <font color="Maroon" title="c3">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c3">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="measure7.html#D12" target="_self" title="MEASURE7:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E9:32_2_7"/>
<font color="Maroon" title="c19">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s2</font>
 <b>by </b><i><a class="txt" href="measure7.html#E1:32"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_7"><i><font color="Green" title="E45">A40</font></i></a>, <a class="txt" href="measure7.html#E4:32_2_7"><i><font color="Green" title="E46">A41</font></i></a>, <a class="txt" href="measure7.html#E6:32_2_7"><i><font color="Green" title="E47">A42</font></i></a>, <a class="txt" href="measure7.html#E7:32_2_7"><i><font color="Green" title="E48">A43</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:32_2"/><b>then </b><i><font color="Green" title="E45">A44</font></i>: 
<font color="Maroon" title="c18">SOS</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/>
<a NAME="E26:32_2"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N0</font> implies <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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">SOS</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">s</font> ) &amp; ( not <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N0</font> implies <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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>thus </b><a NAME="E1:32_2_8"/>
( <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N0</font> implies <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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">SOS</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c19">s</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_8_1"/><i><font color="Green" title="E46">A45</font></i>: 
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N0</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">s1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_2_8_1"/><i><font color="Green" title="E47">A46</font></i>: 
( <font color="Maroon" title="c20">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s1</font> )
 ;<br/>
<a NAME="E4:32_2_8_1"/><i><font color="Green" title="E48">A47</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">SOS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E23:32_2"><i><font color="Green" title="E44">A39</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_8_1"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/>
<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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">s</font> = 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c19">s</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
<br/>.= 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="measure7.html#E3:32_2_8_1"><i><font color="Green" title="E47">A46</font></i></a>, <a class="ref" href="measure7.html#D13" target="_self" title="MEASURE7:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">SOS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E4:32_2_8_1"><i><font color="Green" title="E48">A47</font></i></a>, <a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">SOS</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c19">s</font>
<b>by </b><i><a class="txt" href="measure7.html#E1:32_2_8_1"><i><font color="Green" title="E46">A45</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>
<b>hence </b><a NAME="E6:32_2_8_1"/>
<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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">SOS</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c19">s</font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:32_2_8"/>
not <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N0</font>
 ;<br/>

<a NAME="E3:32_2_8"/><b>then </b><i><font color="Green" title="E46">A48</font></i>: 
not <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">s</font>
 
;<br/>
<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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">s</font> = 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c19">s</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>
<br/>.= 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">s</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="measure7.html#D13" target="_self" title="MEASURE7:def.13">Def13</a></i>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<b>by </b><i><a class="txt" href="measure7.html#E5:32_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="measure7.html#E3:32_2_8"><i><font color="Green" title="E46">A48</font></i></a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a></i>
;<br/>
<b>hence </b><a NAME="E5:32_2_8"/>
<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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E27:32_2"/><b>then </b><i><font color="Green" title="E46">A49</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E14:32_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="txt" href="measure7.html#E17:32_2"><i><font color="Green" title="E41">A36</font></i></a>, <a class="txt" href="measure7.html#E25:32_2"><i><font color="Green" title="E45">A44</font></i></a>, <a class="ref" href="measure7.html#T12" target="_self" title="MEASURE7:th.12">Th12</a></i>;<br/>
<i><font color="Green" title="E47">A50</font></i>:  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> = 
 <a href="measure7.html#K7" title="MEASURE7:func.7">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#D7" target="_self" title="MEASURE7:def.7">Def7</a></i>
;<br/>
<a NAME="E29:32_2"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T14" target="_self" title="MEASURE7:th.14">Th14</a></i>;<br/>
<a NAME="E30:32_2"/><b>then </b><i><font color="Green" title="E48">A51</font></i>: 
 <a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></i>;<br/>
<a NAME="E31:32_2"/><b>then </b><i><font color="Green" title="E49">A52</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#T2" target="_self" title="MEASURE7:th.2">Th2</a></i>;<br/>
<a NAME="E32:32_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">m</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E30:32_2"><i><font color="Green" title="E48">A51</font></i></a>, <a class="ref" href="supinf_2.html#T60" title="SUPINF_2:th.60">SUPINF_2:60</a></i>;<br/>
<a NAME="E33:32_2"/><b>then </b><i><font color="Green" title="E50">A53</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">m</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E28:32_2"><i><font color="Green" title="E47">A50</font></i></a>, <a class="txt" href="measure7.html#E31:32_2"><i><font color="Green" title="E49">A52</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:32_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E16:32_2"><i><font color="Green" title="E40">A35</font></i></a>, <a class="txt" href="measure7.html#E27:32_2"><i><font color="Green" title="E46">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:32_2"/><b>then </b><i><font color="Green" title="E51">A54</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">m</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E33:32_2"><i><font color="Green" title="E50">A53</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E36:32_2"/><i><font color="Green" title="E52">A55</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">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="c5">k</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#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c6">m0</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32_2"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="measure7.html#D13" target="_self" title="MEASURE7:def.13">Def13</a></i>
<br/>.= 
 <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
<b>by </b><i><a class="txt" href="measure7.html#E8:32_2"><i><font color="Green" title="E36">A16</font></i></a></i>
;<br/>
<a NAME="E38:32_2"/><b>then </b><i><font color="Green" title="E53">A56</font></i>: 
<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="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a></i>;<br/>
<a NAME="E39:32_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">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="c5">k</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><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="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63" title="SUPINF_2:th.63">SUPINF_2:63</a></i>;<br/>
<a NAME="E40:32_2"/><b>then </b><i><font color="Green" title="E54">A57</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">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="c5">k</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E38:32_2"><i><font color="Green" title="E53">A56</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/>
<a NAME="E41:32_2"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T14" target="_self" title="MEASURE7:th.14">Th14</a></i>;<br/>
<a NAME="E42:32_2"/><b>then </b>
 <a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></i>;<br/>
<a NAME="E43:32_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c6">m0</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#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">m</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T60" title="SUPINF_2:th.60">SUPINF_2:60</a></i>;<br/>
<a NAME="E44:32_2"/><b>then </b><i><font color="Green" title="E55">A58</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">m</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E36:32_2"><i><font color="Green" title="E52">A55</font></i></a>, <a class="txt" href="measure7.html#E40:32_2"><i><font color="Green" title="E54">A57</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E45:32_2"/>
( <span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="measure7.html#T13" target="_self" title="MEASURE7:th.13">Th13</a></i>;<br/>
<a NAME="E46:32_2"/><b>then </b><i><font color="Green" title="E56">A59</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T59" title="SUPINF_2:th.59">SUPINF_2:59</a></i>;<br/>
<a NAME="E47:32_2"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T14" target="_self" title="MEASURE7:th.14">Th14</a></i>;<br/>
<a NAME="E48:32_2"/><b>then </b><i><font color="Green" title="E57">A60</font></i>: 
 <a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></i>;<br/>
<a NAME="E49:32_2"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T14" target="_self" title="MEASURE7:th.14">Th14</a></i>;<br/>
<a NAME="E50:32_2"/><b>then </b>
 <a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></i>;<br/>
<a NAME="E51:32_2"/><b>then </b><i><font color="Green" title="E58">A61</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c9">G</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c12">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c14">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E13:32_2"><i><font color="Green" title="E37">A21</font></i></a>, <a class="txt" href="measure7.html#E48:32_2"><i><font color="Green" title="E57">A60</font></i></a>, <a class="ref" href="measure7.html#T3" target="_self" title="MEASURE7:th.3">Th3</a></i>;<br/>
<a NAME="E52:32_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <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="c9">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><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="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:32_2_11"/><i><font color="Green" title="E59">A62</font></i>: 
( <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="c9">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span> &amp; <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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span> &amp; <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="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="measure7.html#D4" target="_self" title="MEASURE7:def.4">Def4</a></i>;<br/>
<a NAME="E2:32_2_11"/><b>then </b><i><font color="Green" title="E60">A63</font></i>: 
<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="c9">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="measure7.html#D13" target="_self" title="MEASURE7:def.13">Def13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_11_1"/>
( <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_11_1_1"><b>suppose </b></a><a NAME="E1:32_2_11_1_1"/><i><font color="Green" title="E61">A64</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><i><font color="Green" title="E62">A65</font></i>: <span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c19">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="measure7.html#D13" target="_self" title="MEASURE7:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E5:32_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_11_1_1"><i><font color="Green" title="E61">A64</font></i></a></i>
;<br/><span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c19">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="measure7.html#D13" target="_self" title="MEASURE7:def.13">Def13</a></i>
<br/>.= 
 <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
<b>by </b><i><a class="txt" href="measure7.html#E8:32_2"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_11_1_1"><i><font color="Green" title="E61">A64</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:32_2_11_1_1"/>
<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="c9">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><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="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E1:32_2_11"><i><font color="Green" title="E59">A62</font></i></a>, <a class="txt" href="measure7.html#E2:32_2_11"><i><font color="Green" title="E60">A63</font></i></a>, <a class="txt" href="measure7.html#E2:32_2_11_1_1"><i><font color="Green" title="E62">A65</font></i></a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_11_1_2"><b>suppose </b></a><a NAME="E1:32_2_11_1_2"/><i><font color="Green" title="E61">A66</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><i><font color="Green" title="E62">A67</font></i>: <span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c19">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="measure7.html#D13" target="_self" title="MEASURE7:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E8:32_2"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_11_1_2"><i><font color="Green" title="E61">A66</font></i></a></i>
;<br/><span class="p1">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c19">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">G0</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="measure7.html#D13" target="_self" title="MEASURE7:def.13">Def13</a></i>
<br/>.= 
 <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
<b>by </b><i><a class="txt" href="measure7.html#E5:32_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="measure7.html#E1:32_2_11_1_2"><i><font color="Green" title="E61">A66</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:32_2_11_1_2"/>
<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="c9">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c12">G0</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><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="c14">G1</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="measure7.html#K4" title="MEASURE7:func.4">vol</a> </span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E1:32_2_11"><i><font color="Green" title="E59">A62</font></i></a>, <a class="txt" href="measure7.html#E2:32_2_11"><i><font color="Green" title="E60">A63</font></i></a>, <a class="txt" href="measure7.html#E2:32_2_11_1_2"><i><font color="Green" title="E62">A67</font></i></a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E53:32_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c12">G0</font>,<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E14:32_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="txt" href="measure7.html#E15:32_2"><i><font color="Green" title="E39">A34</font></i></a>, <a class="ref" href="measure7.html#T3" target="_self" title="MEASURE7:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E54:32_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" title="MEASURE7:func.14">On</a> <font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" title="MEASURE7:func.8">vol</a> <font color="Maroon" title="c9">G</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="measure7.html#E35:32_2"><i><font color="Green" title="E51">A54</font></i></a>, <a class="txt" href="measure7.html#E44:32_2"><i><font color="Green" title="E55">A58</font></i></a>, <a class="txt" href="measure7.html#E46:32_2"><i><font color="Green" title="E56">A59</font></i></a>, <a class="txt" href="measure7.html#E51:32_2"><i><font color="Green" title="E58">A61</font></i></a>, <a class="ref" href="measure1.html#T4" title="MEASURE1:th.4">MEASURE1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:32"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="measure7.html#E4:32"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="measure7.html#E5:32"><i><font color="Green" title="E31">A7</font></i></a>);</i><br/>


</div>
