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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Olive" title="b1">F</font><br/> for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b5">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b6">Cvr</font> being   <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b5">Sets</font>,<font color="Olive" title="b1">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Olive" title="b2">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b6">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b6">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b4">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Maroon" title="c2">F</font><br/> for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b4">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b5">Cvr</font> being   <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b4">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Olive" title="b1">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b5">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b5">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">Cvr</font> being   <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b3">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b4">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b4">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">m</font></span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">Cvr</font> being   <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b2">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b3">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b3">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">m</font>;<br/>
<a NAME="E1:24"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">D</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">D</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<a NAME="E4:24"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">a</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">b</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">N0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</font> </span>}</span> ;<br/>
<a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">s1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</font> </span>}</span>  or <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E2:24_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s1</font> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:24_1_1"/>
<font color="Maroon" title="c10">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:24_1"/>
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</font> </span>}</span> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">N0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</font> </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c11">m0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E5:24_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
for <font color="Olive" title="b1">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">Cvr</font> being   <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b1">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b2">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">m0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">take </span><font color="Maroon" title="c12">m</font> = <font color="Maroon" title="c11">m0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">Cvr</font> being   <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b1">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b2">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">Sets</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Cvr</font> being   <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c13">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b1">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c14">Cvr</font> be    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c13">Sets</font>,<font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> $1</span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b1">m</font> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>] <span class="kw">means </span>( ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> $1</span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b1">m</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<a NAME="E6:24_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<div><span class="lab"><font color="Green" title="E21">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c15">n</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/><span class="kw">assume </span><a NAME="E2:24_1_2_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">take </span><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c16">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c17">y</font>]</span><br/><span class="kw">thus </span><a NAME="E3:24_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c17">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:24_1_2"/>
( <font color="Maroon" title="c15">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c5">y</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Olive" title="b1">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c15">Cvr0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:24_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c15">Cvr0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>);<br/></span>
<a NAME="E9:24_1"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c15">Cvr0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> is    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">Cvr0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> is    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">F</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c17">FSets0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:24_1_3"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c15">Cvr0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">FSets0</font>
 <span class="kw">and </span><br/><a NAME="E3:24_1_3"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">FSets0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">FSets0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">FSets0</font> = <font color="Maroon" title="c17">FSets0</font> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_3"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<a NAME="E5:24_1_3"/>
for <font color="Olive" title="b1">s</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c18">FSets0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">s</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">FSets0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E1:24_1_3_1"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_3_1_1"/>
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">FSets0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/><a NAME="E2:24_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">FSets0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E2:24_1_3"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E1:24_1_3_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_1_3_1_1"/>
<font color="Maroon" title="c18">FSets0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:24_1_3_1"/>
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E4:24_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c16">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon" title="c18">FSets0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E2:24_1_3"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E1:24_1_3_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:24_1_3_1"/>
<font color="Maroon" title="c18">FSets0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_3_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:24_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c18">FSets0</font> is    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" target="_self" title="MEASURE8:def.2">Def2</a></span>;<br/>
<a NAME="E7:24_1_3"/>
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E8:24_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E9:24_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">FSets0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:24_1_3"/>
<font color="Maroon" title="c15">Cvr0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> is    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_3"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E6:24_1_3"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="measure8.html#D3" target="_self" title="MEASURE8:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:24_1"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">n</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<div><span class="lab"><font color="Green" title="E24">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c16">n</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/><span class="kw">assume </span><a NAME="E2:24_1_4_1"/><span class="lab"><font color="Green" title="E25">A16</font></span>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">n</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">take </span><font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c17">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c18">y</font>]</span><br/><span class="kw">thus </span><a NAME="E3:24_1_4_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c18">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_4_1"><span class="lab"><font color="Green" title="E25">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:24_1_4"/>
( <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c5">y</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_1_4"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">n</font>,<font color="Olive" title="b1">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_4"><span class="lab"><font color="Green" title="E24">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c16">Cvr1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E12:24_1"/><span class="lab"><font color="Green" title="E24">A17</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c16">Cvr1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E10:24_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a>);<br/></span>
<a NAME="E13:24_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c16">Cvr1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> is    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">Cvr1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font> is    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font>,<font color="Maroon" title="c2">F</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c18">FSets1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:24_1_5"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
<font color="Maroon" title="c16">Cvr1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">FSets1</font>
 <span class="kw">and </span><br/><a NAME="E3:24_1_5"/><span class="lab"><font color="Green" title="E26">A19</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">FSets1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">FSets1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">FSets1</font> = <font color="Maroon" title="c18">FSets1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_5"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<a NAME="E5:24_1_5"/>
for <font color="Olive" title="b1">s</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">FSets1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">s</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">FSets1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E1:24_1_5_1"/><span class="lab"><font color="Green" title="E27">A20</font></span>: 
<font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E2:24_1_5_1"/><span class="lab"><font color="Green" title="E28">A21</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E29">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_5_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_5_1_1"/>
<font color="Maroon" title="c17">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">FSets1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/><a NAME="E2:24_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">FSets1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c20">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c20">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E2:24_1_5"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="txt" href="#E2:24_1_5_1"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="txt" href="#E1:24_1_5_1"><span class="lab"><font color="Green" title="E27">A20</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_1_5_1_1"/>
<font color="Maroon" title="c19">FSets1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:24_1_5_1"/>
( <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon" title="c19">FSets1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c20">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E2:24_1_5"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="txt" href="#E1:24_1_5_1"><span class="lab"><font color="Green" title="E27">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:24_1_5_1"/>
<font color="Maroon" title="c19">FSets1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_5_1"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:24_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A23</font></span>: 
<font color="Maroon" title="c19">FSets1</font> is    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" target="_self" title="MEASURE8:def.2">Def2</a></span>;<br/>
<a NAME="E7:24_1_5"/>
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E8:24_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E9:24_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">FSets1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:24_1_5"/>
<font color="Maroon" title="c16">Cvr1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font> is    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c4">D</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font>,<font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_5"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="txt" href="#E6:24_1_5"><span class="lab"><font color="Green" title="E27">A23</font></span></a>, <a class="ref" href="measure8.html#D3" target="_self" title="MEASURE8:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">Cvr1</font> = <font color="Maroon" title="c16">Cvr1</font> as    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D4" target="_self" title="MEASURE8:def.4">Def4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">Cvr0</font> = <font color="Maroon" title="c15">Cvr0</font> as    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="measure8.html#D4" target="_self" title="MEASURE8:def.4">Def4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">PSv1</font> =  <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D10" target="_self" title="MEASURE8:def.10">Def10</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E24">A17</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E25">A24</font></span>: <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">PSv0</font> =  <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">PSv</font> =  <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c10">N0</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1;<br/>
<a NAME="E18:24_1"/><span class="lab"><font color="Green" title="E26">A25</font></span>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c10">N0</font>  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">SOS</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c10">N0</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E20:24_1"/><span class="lab"><font color="Green" title="E27">A26</font></span>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c10">N0</font> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E18:24_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a>);<br/></span>
<a NAME="E21:24_1"/><span class="lab"><font color="Green" title="E28">A27</font></span>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> implies <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">SOS</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> ) &amp; ( not <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> implies <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c23">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> implies <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">SOS</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> ) &amp; ( not <font color="Maroon" title="c23">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> implies <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:24_1_8"/>
( <font color="Maroon" title="c23">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> implies <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">SOS</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c23">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> implies <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_8_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:24_1_8_1"/><span class="lab"><font color="Green" title="E29">A28</font></span>: 
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:24_1_8_1"/><span class="lab"><font color="Green" title="E30">A29</font></span>: 
<font color="Maroon" title="c23">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">SOS</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font></span><br/>

<a NAME="E3:24_1_8_1"/><span class="kw">then </span>
(  ex <font color="Olive" title="b1">s1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s</font> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s1</font> ) &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:24_1"><span class="lab"><font color="Green" title="E27">A26</font></span></a></span>;<br/>
<a NAME="E4:24_1_8_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_8_1"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="ref" href="measure8.html#D10" target="_self" title="MEASURE8:def.10">Def10</a></span>;<br/>
<a NAME="E5:24_1_8_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:24_1_8_1"/>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">SOS</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_8_1"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:24_1_8"/>
not <font color="Maroon" title="c23">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E3:24_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A30</font></span>: 
not <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s</font>
 
;<br/>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> = 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D10" target="_self" title="MEASURE8:def.10">Def10</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E3:24_1_8"><span class="lab"><font color="Green" title="E29">A30</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:24_1_8"/>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">s1</font>, <font color="Maroon" title="c24">s2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> &amp; <font color="Maroon" title="c24">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> &amp; <font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">s2</font> implies <font color="Maroon" title="c23">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24_1_9"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
( <font color="Maroon" title="c23">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> &amp; <font color="Maroon" title="c24">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N0</font> )
 <span class="kw">and </span><br/><a NAME="E2:24_1_9"/><span class="lab"><font color="Green" title="E30">A32</font></span>: 
<font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s2</font></span><br/><a NAME="E3:24_1_9"/><span class="lab"><font color="Green" title="E31">A33</font></span>: 
(  ex <font color="Olive" title="b1">s11</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">s11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s1</font> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s11</font> ) &amp;  ex <font color="Olive" title="b1">s22</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">s22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s2</font> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s22</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_9"><span class="lab"><font color="Green" title="E29">A31</font></span></a></span>;<br/><a NAME="E4:24_1_9"/><span class="lab"><font color="Green" title="E32">A34</font></span>: 
( <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">s2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">s2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_9"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="ref" href="funct_2.html#T196" title="FUNCT_2:th.196">FUNCT_2:196</a></span>;<br/><a NAME="E5:24_1_9"/>
( <font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">s1</font> &amp; <font color="Maroon" title="c22">SOS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">s2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:24_1"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E1:24_1_9"><span class="lab"><font color="Green" title="E29">A31</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:24_1_9"/>
<font color="Maroon" title="c23">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_9"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E3:24_1_9"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E4:24_1_9"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a>, <a class="ref" href="nagata_2.html#T2" title="NAGATA_2:th.2">NAGATA_2:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E23:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A35</font></span>: 
<font color="Maroon" title="c22">SOS</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></span>;<br/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T63" title="SUPINF_2:th.63">SUPINF_2:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24_1"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>
;<br/>
<a NAME="E25:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<a NAME="E26:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A36</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<a NAME="E27:24_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T5" target="_self" title="MEASURE8:th.5">Th5</a></span>;<br/>
<a NAME="E28:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A37</font></span>: 
 <a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></span>;<br/>
<a NAME="E29:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#T2" title="MEASURE7:th.2">MEASURE7:2</a></span>;<br/>
<a NAME="E30:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A38</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D6" target="_self" title="MEASURE8:def.6">Def6</a></span>;<br/>
<a NAME="E31:24_1"/><span class="lab"><font color="Green" title="E33">A39</font></span>: 
<font color="Maroon" title="c11">m0</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E32:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:24_1"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="ref" href="supinf_2.html#T60" title="SUPINF_2:th.60">SUPINF_2:60</a></span>;<br/>
<a NAME="E33:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A40</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:24_1"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E34:24_1"/>
 <a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T4" target="_self" title="MEASURE8:th.4">Th4</a></span>;<br/>
<a NAME="E35:24_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:24_1"><span class="lab"><font color="Green" title="E29">A35</font></span></a>, <a class="txt" href="#E21:24_1"><span class="lab"><font color="Green" title="E28">A27</font></span></a>, <a class="ref" href="measure7.html#T12" title="MEASURE7:th.12">MEASURE7:12</a></span>;<br/>
<a NAME="E36:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A41</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:24_1"><span class="lab"><font color="Green" title="E34">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E37:24_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T4" target="_self" title="MEASURE8:th.4">Th4</a>, <a class="ref" href="measure7.html#T7" title="MEASURE7:th.7">MEASURE7:7</a></span>;<br/>
<a NAME="E38:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:24_1"><span class="lab"><font color="Green" title="E35">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E39:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<a NAME="E40:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A42</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<a NAME="E41:24_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T5" target="_self" title="MEASURE8:th.5">Th5</a></span>;<br/>
<a NAME="E42:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A43</font></span>: 
 <a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></span>;<br/>
<a NAME="E43:24_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> &amp;  <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc9.html#T16" title="MESFUNC9:th.16">MESFUNC9:16</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E44:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A44</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">m0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:24_1"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="ref" href="rinfsup2.html#T7" title="RINFSUP2:th.7">RINFSUP2:7</a></span>;<br/>
<a NAME="E45:24_1"/><span class="lab"><font color="Green" title="E39">A45</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c24">n1</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">n2</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font>;<br/>
<a NAME="E1:24_1_11"/><span class="lab"><font color="Green" title="E40">A46</font></span>: 
( <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>;<br/>
<a NAME="E2:24_1_11"/>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>;<br/>
<a NAME="E3:24_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A47</font></span>: 
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D10" target="_self" title="MEASURE8:def.10">Def10</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_11_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1_11_1"/>
( <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_11_1_1"/><span class="lab"><font color="Green" title="E42">A48</font></span>: 
<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span><br/><div class="add"><span class="p1">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D10" target="_self" title="MEASURE8:def.10">Def10</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E1:24_1_11_1_1"><span class="lab"><font color="Green" title="E42">A48</font></span></a></span>
;<br/><a NAME="E3:24_1_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A49</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/><span class="p1">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D10" target="_self" title="MEASURE8:def.10">Def10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E1:24_1_11_1_1"><span class="lab"><font color="Green" title="E42">A48</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:24_1_11_1_1"/>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_11"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="txt" href="#E3:24_1_11"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="txt" href="#E3:24_1_11_1_1"><span class="lab"><font color="Green" title="E43">A49</font></span></a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_11_1_2"/><span class="lab"><font color="Green" title="E42">A50</font></span>: 
<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span><br/><div class="add"><span class="p1">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D10" target="_self" title="MEASURE8:def.10">Def10</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E1:24_1_11_1_2"><span class="lab"><font color="Green" title="E42">A50</font></span></a></span>
;<br/><a NAME="E3:24_1_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A51</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/><span class="p1">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D10" target="_self" title="MEASURE8:def.10">Def10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E1:24_1_11_1_2"><span class="lab"><font color="Green" title="E42">A50</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:24_1_11_1_2"/>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_11"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="txt" href="#E3:24_1_11"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="txt" href="#E3:24_1_11_1_2"><span class="lab"><font color="Green" title="E43">A51</font></span></a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E46:24_1"/>
(  <a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp;  <a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T4" target="_self" title="MEASURE8:th.4">Th4</a></span>;<br/>
<a NAME="E47:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:24_1"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="ref" href="measure7.html#T3" title="MEASURE7:th.3">MEASURE7:3</a></span>;<br/>
<a NAME="E48:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<a NAME="E49:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<a NAME="E50:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A52</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<a NAME="E51:24_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">m0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>;<br/>
<a NAME="E52:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A53</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:24_1"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="txt" href="#E44:24_1"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E53:24_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c14">Cvr</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c14">Cvr</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E42">A54</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_12_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_12_1"/><span class="lab"><font color="Green" title="E43">A55</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span></span>)</span></span><br/><a NAME="E2:24_1_12_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_12_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><br/>

<a NAME="E1:24_1_12_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E1:24_1_12_1"><span class="lab"><font color="Green" title="E43">A55</font></span></a></span>;<br/>
<a NAME="E2:24_1_12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c24">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_1_12_1_1"/>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:24_1_12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A56</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#T1" title="MEASURE7:th.1">MEASURE7:1</a></span>;<br/><a NAME="E4:24_1_12_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_12_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> )</span><br/>

<a NAME="E1:24_1_12_1_2"/>
( <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c24">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c24">s</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:24_1_12_1_2"/>
( <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E1:24_1_12_1"><span class="lab"><font color="Green" title="E43">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:24_1_12_1"/><span class="kw">then </span>
(  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> &amp;  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></span>;<br/><a NAME="E6:24_1_12_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:24_1_12_1"/>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_12_1"><span class="lab"><font color="Green" title="E44">A56</font></span></a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E43">A57</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_12_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_12_2"/><span class="lab"><font color="Green" title="E44">A58</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span></span>)</span></span><br/><a NAME="E2:24_1_12_2"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_12_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E1:24_1_12_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E1:24_1_12_2"><span class="lab"><font color="Green" title="E44">A58</font></span></a></span>;<br/>
<a NAME="E2:24_1_12_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c24">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_1_12_2_1"/>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:24_1_12_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A59</font></span>: 
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#T1" title="MEASURE7:th.1">MEASURE7:1</a></span>;<br/><a NAME="E4:24_1_12_2"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">s</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_12_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> )</span><br/>

<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> = 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c24">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c24">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E1:24_1_12_2"><span class="lab"><font color="Green" title="E44">A58</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:24_1_12_2_2"/>
( <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c24">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:24_1_12_2"/><span class="kw">then </span>
(  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> &amp;  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></span>;<br/><a NAME="E6:24_1_12_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:24_1_12_2"/>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c17">Cvr1</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_12_2"><span class="lab"><font color="Green" title="E45">A59</font></span></a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:24_1_12"/>
( <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c14">Cvr</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">Cvr0</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D6" target="_self" title="MEASURE8:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:24_1_12"/>
<span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c14">Cvr</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_12"><span class="lab"><font color="Green" title="E43">A57</font></span></a>, <a class="txt" href="#E1:24_1_12"><span class="lab"><font color="Green" title="E42">A54</font></span></a>, <a class="ref" href="measure8.html#D6" target="_self" title="MEASURE8:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E54:24_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:24_1"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="txt" href="#E42:24_1"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="ref" href="measure7.html#T3" title="MEASURE7:th.3">MEASURE7:3</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p3">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c18">Cvr0</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c17">Cvr1</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E56:24_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c14">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:24_1"><span class="lab"><font color="Green" title="E36">A42</font></span></a>, <a class="txt" href="#E52:24_1"><span class="lab"><font color="Green" title="E41">A53</font></span></a>, <a class="txt" href="#E50:24_1"><span class="lab"><font color="Green" title="E40">A52</font></span></a>, <a class="ref" href="xxreal_3.html#T38" title="XXREAL_3:th.38">XXREAL_3:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:24"/><span class="lab"><font color="Green" title="E18">A60</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c6">m</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Sets</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">Cvr</font> being   <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Olive" title="b1">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b2">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">Sets</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Cvr</font> being   <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c7">Sets</font>,<font color="Maroon" title="c2">F</font> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Olive" title="b1">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">Cvr</font> be    <a href="measure8.html#M4" title="MEASURE8:mode.4">Covering</a> of <font color="Maroon" title="c7">Sets</font>,<font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font></span><br/>



<a NAME="E1:24_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">Cvr</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T5" target="_self" title="MEASURE8:th.5">Th5</a></span>;<br/>
<a NAME="E2:24_2"/><span class="kw">then </span>
 <a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">Cvr</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></span>;<br/>
<a NAME="E3:24_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">Cvr</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#T2" title="MEASURE7:th.2">MEASURE7:2</a></span>;<br/>
<a NAME="E4:24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A61</font></span>: 
<span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">Cvr</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>;<br/>
<a NAME="E5:24_2"/>
( <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c8">Cvr</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>;<br/>
<a NAME="E6:24_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D10" target="_self" title="MEASURE8:def.10">Def10</a></span>;<br/>
<a NAME="E7:24_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T4" target="_self" title="MEASURE8:th.4">Th4</a>, <a class="ref" href="measure6.html#T3" title="MEASURE6:th.3">MEASURE6:3</a></span>;<br/>
<a NAME="E8:24_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">Cvr</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D6" target="_self" title="MEASURE8:def.6">Def6</a>, <a class="ref" href="mesfunc9.html#D1" title="MESFUNC9:def.1">MESFUNC9:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:24_2"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><a href="measure8.html#K8" title="MEASURE8:func.8">On</a> <font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="measure8.html#K4" title="MEASURE8:func.4">Volume</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">Cvr</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_2"><span class="lab"><font color="Green" title="E19">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E6:24"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E18">A60</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
