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

<b>let </b><font color="Maroon">H</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">H</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E27</font></i>: 
 <a href="funcop_1.html#K9">rng</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">D</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> <br/> for <font color="Olive">G</font> being   <a href="measure7.html#M2" target="_self">Interval_Covering</a> of <font color="Olive">F</font> holds  <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Olive">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Olive">G</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">m</font>;<br/>
<a NAME="E4:32"/><i><font color="Green">E28</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">m</font> = <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">G</font> be    <a href="measure7.html#M2" target="_self">Interval_Covering</a> of <font color="Maroon">F</font>;<br/>



<b>reconsider </b><font color="Maroon">GG</font> =  <a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> as    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<a NAME="E2:32_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">GG</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">GG</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> 0
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E3:32_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">GG</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><font color="Maroon">GG</font> <a href="measure7.html#K3" target="_self">.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:32_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">GG</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32_1"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E5:32_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:32_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">GG</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E8:32_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="measure6.html#T3">MEASURE6:3</a></i>;<br/>
<a NAME="E7:32_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">GG</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:32_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">GG</font> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:32_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E10:32_1"/><b>then </b>
 <a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E11:32_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:32_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="measure7.html#E4:32_1"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">m0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:32_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> <br/> for <font color="Olive">G</font> being   <a href="measure7.html#M2" target="_self">Interval_Covering</a> of <font color="Olive">F</font> holds  <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Olive">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Olive">G</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m0</font>
 ;<br/>

<b>take </b><font color="Maroon">m</font> = <font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">G</font> be    <a href="measure7.html#M2" target="_self">Interval_Covering</a> of <font color="Maroon">F</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Olive">m</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  ) );<br/>
<a NAME="E2:32_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</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">E43</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><b>take </b>
<font color="Maroon">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">n</font>,<font color="Maroon">y</font>]
 <b>by </b><i><a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</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">E45</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">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">n</font>,<font color="Maroon">y</font>]
 <b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green">E27</font></i></a>, <a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G0</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span><b> such that </b><br/><a NAME="E4:32_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Olive">m</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  ) );<br/>
<a NAME="E5:32_2"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</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">E60</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><b>take </b>
<font color="Maroon">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">n</font>,<font color="Maroon">y</font>]
 <b>by </b><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">E61</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">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">n</font>,<font color="Maroon">y</font>]
 <b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green">E27</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="funct_2.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> </span>)</span><b> such that </b><br/><a NAME="E7:32_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="measure7.html#T2" target="_self">Th2</a>);<br/></i>
<a NAME="E8:32_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> is    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">f0</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:32_2_3"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f0</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f0</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f0</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f0</font> = <font color="Maroon">f0</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="measure7.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E4:32_2_3"/>
<font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E5:32_2_3"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E6:32_2_3"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f0</font> <a href="prob_1.html#K1">.</a> <font color="Olive">s</font> is   <a href="measure5.html#NM1">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">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</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">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:32_2_3_1_1_1"/><b>then </b>
<font color="Maroon">f0</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="ref" href="measure7.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E3:32_2_3_1_1_1"/>
<font color="Maroon">f0</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">s</font> is   <a href="measure5.html#NM1">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">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_3_1_1_2"/>
<font color="Maroon">f0</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">s</font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i><a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="ref" href="measure7.html#T3" target="_self">Th3</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">f0</font> = <font color="Maroon">f0</font> as    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <b>by </b><i>, </i>;<br/>
<a NAME="E8:32_2_3"/>
<font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f0</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E9:32_2_3"/>
<font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G0</font> = <font color="Maroon">G0</font> as    <a href="measure7.html#M2" target="_self">Interval_Covering</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">GG0</font> =  <a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font>;<br/>
<a NAME="E10:32_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> is    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">f0</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:32_2_4"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f0</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f0</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f0</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f0</font> = <font color="Maroon">f0</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="measure7.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E4:32_2_4"/>
<font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E5:32_2_4"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E6:32_2_4"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f0</font> <a href="prob_1.html#K1">.</a> <font color="Olive">s</font> is   <a href="measure5.html#NM1">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">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</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">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:32_2_4_1_1_1"/><b>then </b>
<font color="Maroon">f0</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="measure7.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E3:32_2_4_1_1_1"/>
<font color="Maroon">f0</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">s</font> is   <a href="measure5.html#NM1">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">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_4_1_1_2"/>
<font color="Maroon">f0</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">s</font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i>, <a class="ref" href="measure7.html#T3" target="_self">Th3</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">f0</font> = <font color="Maroon">f0</font> as    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <b>by </b><i>, </i>;<br/>
<a NAME="E8:32_2_4"/>
<font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f0</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E9:32_2_4"/>
<font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">D</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G1</font> as    <a href="measure7.html#M2" target="_self">Interval_Covering</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">GG1</font> =  <a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font>;<br/>
<a NAME="E12:32_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:32_2_5"/><i><font color="Green">E75</font></i>: 
( <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:32_2_5"/>
 <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" target="_self">vol</a> <span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" target="_self">vol</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">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">E76</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><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">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</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">E77</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:32_2_5_1_1_1"/><i><font color="Green">E78</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">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">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:32_2_5_1_1_1_1_1"/>
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i>, <a class="txt" href="measure7.html#E1:32_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="measure5.html#T60">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">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_2_5_1_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:32_2_5_1_1_1_1"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> = 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="measure7.html#E1:32_2"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><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">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">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">E81</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T62">SUPINF_2:62</a></i>;<br/>
<a NAME="E5:32_2_5_1_1_1_1"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> = 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="measure7.html#E1:32_2"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><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">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">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#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T62">SUPINF_2:62</a></i>;<br/>
<a NAME="E7:32_2_5_1_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E8:32_2_5_1_1_1_1"/>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:32_2_5_1_1_1"/>
 <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" target="_self">vol</a> <span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" target="_self">vol</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E2:32_2"><i><font color="Green">E42</font></i></a></i>;<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">E82</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:32_2_5_1_1_2"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span> )
 
<b>by </b><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">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="measure7.html#E1:32_2"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_2_5_1_1_2"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:32_2_5_1_1_2_2"/>
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="measure7.html#E1:32_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="measure5.html#T60">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">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:32_2_5_1_1_2"/><i><font color="Green">E85</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">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">E87</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:32_2_5_1_1_2_3"/><i><font color="Green">E88</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:32_2_5_1_1_2_3"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32_2"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:32_2_5_1_1_2_3"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T62">SUPINF_2:62</a></i>;<br/>
<a NAME="E5:32_2_5_1_1_2_3"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font>
 
<b>by </b><i><a class="txt" href="measure7.html#E2:32_2"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E6:32_2_5_1_1_2_3"/><b>then </b>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T62">SUPINF_2:62</a></i>;<br/>
<a NAME="E7:32_2_5_1_1_2_3"/><b>then </b>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E8:32_2_5_1_1_2_3"/>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19">SUM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E4:32_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:32_2_5_1_1_2"/>
 <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" target="_self">vol</a> <span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="measure7.html#K7" target="_self">vol</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="measure7.html#T4" target="_self">Th4</a></i>;<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" target="_self">vol</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_2"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E14:32_2"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E15:32_2"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:32_2"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">N0</font> = <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> </span>}</span> ;<br/>
<a NAME="E17:32_2"/><i><font color="Green">E94</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">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">s1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_2_6"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> </span>}</span> 
 ;<br/>

<a NAME="E2:32_2_6"/><b>then </b>
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> )
 
;<br/>
<b>hence </b><a NAME="E3:32_2_6"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:32_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">N0</font> = <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">N0</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E20:32_2"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">N0</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">s</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">SOS</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N0</font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E22:32_2"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">N0</font> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">s</font>,<font color="Maroon">SOS</font> <a href="measure7.html#K1" target="_self">.</a> <font color="Olive">s</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="measure7.html#T4" target="_self">Th4</a>);<br/></i>
<a NAME="E23:32_2"/>
for <font color="Olive">s1</font>, <font color="Olive">s2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> &amp; <font color="Olive">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> &amp; <font color="Maroon">SOS</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SOS</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> holds <br/><font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">s1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">s2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_2_7"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> &amp; <font color="Maroon">SOS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SOS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s1</font>, <font color="Maroon">s2</font> = <font color="Maroon">s2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>consider </b><font color="Maroon">s11</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:32_2_7"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">s11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s11</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E4:32_2"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">s22</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:32_2_7"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">s22</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s22</font> )
 <b>by </b><i><a class="txt" href="measure7.html#E4:32_2"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E7:32_2_7"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">SOS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">SOS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font> )
 
<b>by </b><i>, <a class="txt" href="measure7.html#E4:32_2"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E8:32_2_7"/>
( <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="txt" href="measure7.html#E1:32"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:32_2_7"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, <a class="txt" href="measure7.html#E4:32_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="measure7.html#T6" target="_self">Th6</a>, , <a class="ref" href="measure7.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:32_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">SOS</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<a NAME="E25:32_2"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">SOS</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">s</font> ) &amp; ( not <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><a NAME="E1:32_2_8"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">SOS</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">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">E102</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">s1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_2_8_1"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font> )
 ;<br/>
<a NAME="E4:32_2_8_1"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SOS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="measure7.html#T6" target="_self">Th6</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> = 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">SOS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">SOS</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="measure7.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_2.html#T21">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" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">SOS</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">s</font>
 ;<br/>


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

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

<a NAME="E3:32_2_8"/><b>then </b><i><font color="Green">E106</font></i>: 
not <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> = 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="ref" href="measure7.html#T6" target="_self">Th6</a>, <a class="ref" href="measure5.html#T60">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" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E26:32_2"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i>, <a class="txt" href="measure7.html#E2:32_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="measure7.html#E4:32_2"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<i><font color="Green">E108</font></i>:  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> = 
 <a href="measure7.html#K7" target="_self">vol</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E28:32_2"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E29:32_2"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E30:32_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E31:32_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T7" target="_self">Th7</a>, <a class="ref" href="supinf_2.html#T60">SUPINF_2:60</a></i>;<br/>
<a NAME="E32:32_2"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E33:32_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E1:32_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="measure7.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:32_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="measure7.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:32_2"/><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m0</font>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="measure5.html#K6">{}</a> 
<b>by </b><i/>
;<br/>
<a NAME="E37:32_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i>, <a class="ref" href="measure5.html#T60">MEASURE5:60</a></i>;<br/>
<a NAME="E38:32_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E39:32_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<a NAME="E40:32_2"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E41:32_2"/><b>then </b>
 <a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E42:32_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m0</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T60">SUPINF_2:60</a></i>;<br/>
<a NAME="E43:32_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="measure7.html#T8" target="_self">Th8</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E44:32_2"/>
( <span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a> &amp; <span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a>  is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E45:32_2"/><b>then </b><i><font color="Green">E117</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/>
<a NAME="E46:32_2"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E47:32_2"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E48:32_2"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">s</font>
 
<b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E49:32_2"/><b>then </b>
 <a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E50:32_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G0</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure7.html#T3" target="_self">Th3</a>, <a class="ref" href="measure7.html#T10" target="_self">Th10</a>, </i>;<br/>
<a NAME="E51:32_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:32_2_11"/><i><font color="Green">E120</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:32_2_11"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K5">vol</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, </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" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</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">E159</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><i><font color="Green">E160</font></i>: <span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="measure7.html#E7:32_2"><i><font color="Green">E62</font></i></a></i>
;<br/><span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="measure5.html#K6">{}</a> 
<b>by </b><i>, <a class="txt" href="measure7.html#E7:32_2"><i><font color="Green">E62</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" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="measure7.html#T11" target="_self">Th11</a>, <a class="ref" href="measure5.html#T60">MEASURE5:60</a>, <a class="ref" href="supinf_2.html#T18">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">E161</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><i><font color="Green">E162</font></i>: <span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="measure7.html#E7:32_2"><i><font color="Green">E62</font></i></a></i>
;<br/><span class="p1">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="measure5.html#K6">{}</a> 
<b>by </b><i><a class="txt" href="measure7.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="measure7.html#E7:32_2"><i><font color="Green">E62</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" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="measure7.html#T11" target="_self">Th11</a>, <a class="ref" href="measure5.html#T60">MEASURE5:60</a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E52:32_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G0</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G1</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E53:32_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K14" target="_self">On</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="measure7.html#K4" target="_self">vol</a> </span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K8" target="_self">vol</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="measure7.html#T9" target="_self">Th9</a>, <a class="txt" href="measure7.html#E5:32_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="measure1.html#T4">MEASURE1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:32"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div>
