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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green" title="E91">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc2.html#R2" title="MESFUNC2:pred.2">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green" title="E92">A2</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>and </b><br/><a NAME="E3:97"/><i><font color="Green" title="E93">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">B</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font>;<br/>
<b>set </b><font color="Maroon" title="c9">g2</font> = <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font>;<br/>
<a NAME="E4:97"/><i><font color="Green" title="E94">A4</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc2.html#R2" title="MESFUNC2:pred.2">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97"><i><font color="Green" title="E91">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T40" target="_self" title="MESFUNC5:th.40">Th40</a></i>;<br/>
<a NAME="E5:97"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:97"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/>
<a NAME="E6:97"/><b>then </b><i><font color="Green" title="E95">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">G</font> being   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c11">b</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E8:97"/><i><font color="Green" title="E96">A6</font></i>: 
( <font color="Maroon" title="c10">G</font>,<font color="Maroon" title="c11">b</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> &amp; <font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">b</font> holds <br/>( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:97"><i><font color="Green" title="E94">A4</font></i></a>, <a class="ref" href="mesfunc3.html#T14" title="MESFUNC3:th.14">MESFUNC3:14</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> $1</span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c10">G</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E10:97"/><i><font color="Green" title="E97">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font> &amp; ( for <font color="Olive" title="b1">j</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="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font></span>)</span> holds <br/><font color="Maroon" title="c12">y</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<i><font color="Green" title="E98">A8</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">y</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="mesfunc5.html#E10:97"><i><font color="Green" title="E97">A7</font></i></a></i>
;<br/>
<a NAME="E12:97"/><i><font color="Green" title="E99">A9</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font> <a href="mesfunc2.html#R2" title="MESFUNC2:pred.2">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97"><i><font color="Green" title="E91">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T40" target="_self" title="MESFUNC5:th.40">Th40</a></i>;<br/>
<a NAME="E13:97"/><i><font color="Green" title="E100">A10</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font> <a href="mesfunc2.html#R2" title="MESFUNC2:pred.2">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97"><i><font color="Green" title="E91">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T40" target="_self" title="MESFUNC5:th.40">Th40</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font>;<br/>
<b>consider </b><font color="Maroon" title="c13">G1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E15:97"/><i><font color="Green" title="E101">A11</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font> &amp; ( 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="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font></span>)</span> holds <br/><font color="Maroon" title="c13">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<i><font color="Green" title="E102">A12</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:97"><i><font color="Green" title="E101">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E17:97"/><i><font color="Green" title="E103">A13</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font> holds <br/><font color="Maroon" title="c13">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_3"/>
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font>
 ;<br/>

<a NAME="E2:97_3"/><b>then </b>
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:97_3"/>
<font color="Maroon" title="c13">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:97"><i><font color="Green" title="E101">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">G1</font> = <font color="Maroon" title="c13">G1</font> as   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E16:97"><i><font color="Green" title="E102">A12</font></i></a>, <a class="ref" href="mesfunc5.html#T41" target="_self" title="MESFUNC5:th.41">Th41</a></i>;<br/>
<i><font color="Green" title="E104">A14</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E20:97"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_5"/><i><font color="Green" title="E105">A15</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
 ;<br/>

<a NAME="E2:97_5"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:97_5"/><b>then </b><i><font color="Green" title="E106">A16</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_5"><i><font color="Green" title="E105">A15</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_5"><i><font color="Green" title="E106">A16</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/>
<b>hence </b><a NAME="E5:97_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E19:97"><i><font color="Green" title="E104">A14</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_5"><i><font color="Green" title="E105">A15</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:97"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:97"><i><font color="Green" title="E104">A14</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E22:97"/><b>then </b><i><font color="Green" title="E105">A17</font></i>: 
<font color="Maroon" title="c14">G1</font>,<font color="Maroon" title="c11">b</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E16:97"><i><font color="Green" title="E102">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E17:97"><i><font color="Green" title="E103">A13</font></i></a>, <a class="ref" href="mesfunc5.html#T42" target="_self" title="MESFUNC5:th.42">Th42</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>;<br/>
<b>consider </b><font color="Maroon" title="c15">G2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E24:97"/><i><font color="Green" title="E106">A18</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font> &amp; ( 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="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font></span>)</span> holds <br/><font color="Maroon" title="c15">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<i><font color="Green" title="E107">A19</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">G2</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E24:97"><i><font color="Green" title="E106">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">G2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E26:97"/><i><font color="Green" title="E108">A20</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font> holds <br/><font color="Maroon" title="c15">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_7"/>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font>
 ;<br/>

<a NAME="E2:97_7"/><b>then </b>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:97_7"/>
<font color="Maroon" title="c15">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E24:97"><i><font color="Green" title="E106">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c16">G2</font> = <font color="Maroon" title="c15">G2</font> as   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E25:97"><i><font color="Green" title="E107">A19</font></i></a>, <a class="ref" href="mesfunc5.html#T41" target="_self" title="MESFUNC5:th.41">Th41</a></i>;<br/>
<i><font color="Green" title="E109">A21</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E29:97"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_9"/><i><font color="Green" title="E110">A22</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span>
 ;<br/>

<a NAME="E2:97_9"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:97_9"/><b>then </b><i><font color="Green" title="E111">A23</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_9"><i><font color="Green" title="E110">A22</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_9"><i><font color="Green" title="E111">A23</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/>
<b>hence </b><a NAME="E5:97_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E28:97"><i><font color="Green" title="E109">A21</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_9"><i><font color="Green" title="E110">A22</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:97"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:97"><i><font color="Green" title="E109">A21</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E31:97"/><b>then </b><i><font color="Green" title="E110">A24</font></i>: 
<font color="Maroon" title="c16">G2</font>,<font color="Maroon" title="c11">b</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E25:97"><i><font color="Green" title="E107">A19</font></i></a>, <a class="txt" href="mesfunc5.html#E26:97"><i><font color="Green" title="E108">A20</font></i></a>, <a class="ref" href="mesfunc5.html#T42" target="_self" title="MESFUNC5:th.42">Th42</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:97_10"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_1"><b>suppose </b></a><a NAME="E1:97_10_1"/><i><font color="Green" title="E111">A25</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_1"/><b>then </b><i><font color="Green" title="E112">A26</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/><a NAME="E3:97_10_1"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:97_10_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E5:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E6:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E7:97_10_1"/><b>then </b><i><font color="Green" title="E113">A27</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_1"><i><font color="Green" title="E111">A25</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E8:97_10_1"/><i><font color="Green" title="E114">A28</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:97_10_1"><i><font color="Green" title="E113">A27</font></i></a>, <a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/><a NAME="E9:97_10_1"/>
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E10:97_10_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E11:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E12:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E13:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_1"><i><font color="Green" title="E111">A25</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E14:97_10_1"/><b>then </b>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E15:97_10_1"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:97_10_1"><i><font color="Green" title="E112">A26</font></i></a>, <a class="txt" href="mesfunc5.html#E8:97_10_1"><i><font color="Green" title="E114">A28</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2"><b>suppose </b></a><a NAME="E1:97_10_2"/><i><font color="Green" title="E111">A29</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">y</font> holds <br/><font color="Maroon" title="c12">y</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c10">G</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_10_2_1"/>
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">y</font>
 ;<br/>

<a NAME="E2:97_10_2_1"/><b>then </b>
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:97_10_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c10">G</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:97"><i><font color="Green" title="E97">A7</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:97_10_2"/><b>then </b>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:97"><i><font color="Green" title="E94">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E6:97"><i><font color="Green" title="E95">A5</font></i></a>, <a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E11:97"><i><font color="Green" title="E98">A8</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2"><i><font color="Green" title="E111">A29</font></i></a>, <a class="ref" href="mesfunc4.html#T3" title="MESFUNC4:th.3">MESFUNC4:3</a></i>;<br/><a NAME="E4:97_10_2"/><b>then </b><i><font color="Green" title="E112">A30</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2"><i><font color="Green" title="E111">A29</font></i></a>, <a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:97_10_2_2"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_1"><b>suppose </b></a><a NAME="E1:97_10_2_2_1"/><i><font color="Green" title="E113">A31</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_2_2_1"/><i><font color="Green" title="E114">A32</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_1"><i><font color="Green" title="E113">A31</font></i></a>, <a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/><i><font color="Green" title="E115">A33</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_1"><i><font color="Green" title="E113">A31</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:97_10_2_2_1_2"/><i><font color="Green" title="E116">A34</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:97_10_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E3:97_10_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_1"><i><font color="Green" title="E115">A33</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_1_2"><i><font color="Green" title="E116">A34</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div><a NAME="E5:97_10_2_2_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_1"><i><font color="Green" title="E115">A33</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E6:97_10_2_2_1"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:97_10_2_2_1"><i><font color="Green" title="E114">A32</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2"><b>suppose </b></a><a NAME="E1:97_10_2_2_2"/><i><font color="Green" title="E113">A35</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:97_10_2_2_2_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:97_10_2_2_2_1_1"/><i><font color="Green" title="E114">A36</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_2_2_2_1_1"/><i><font color="Green" title="E115">A37</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_1"><i><font color="Green" title="E114">A36</font></i></a>, <a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/><i><font color="Green" title="E116">A38</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_1"><i><font color="Green" title="E114">A36</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:97_10_2_2_2_1_1_2"/><i><font color="Green" title="E117">A39</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:97_10_2_2_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E3:97_10_2_2_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_1"><i><font color="Green" title="E116">A38</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_1_2"><i><font color="Green" title="E117">A39</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div><a NAME="E5:97_10_2_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_1"><i><font color="Green" title="E116">A38</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E6:97_10_2_2_2_1_1"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:97_10_2_2_2_1_1"><i><font color="Green" title="E115">A37</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:97_10_2_2_2_1_2"/><i><font color="Green" title="E114">A40</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_2_2_2_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:97"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/><a NAME="E3:97_10_2_2_2_1_2"/><b>then </b><i><font color="Green" title="E115">A41</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><a NAME="E4:97_10_2_2_2_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:97"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/><a NAME="E5:97_10_2_2_2_1_2"/><b>then </b><i><font color="Green" title="E116">A42</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> $1</span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c17">y1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E7:97_10_2_2_2_1_2"/><i><font color="Green" title="E117">A43</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">G1</font> &amp; ( for <font color="Olive" title="b1">j</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="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">G1</font></span>)</span> holds <br/><font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><i><font color="Green" title="E118">A44</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:97_10_2_2_2_1_2_2"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font>
 ;<br/><a NAME="E2:97_10_2_2_2_1_2_2"/><b>then </b>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">G1</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:97_10_2_2_2_1_2"><i><font color="Green" title="E118">A44</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:97_10_2_2_2_1_2_2"/>
<font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:97_10_2_2_2_1_2"/><b>then </b>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c17">y1</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:97"><i><font color="Green" title="E99">A9</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2"><i><font color="Green" title="E113">A35</font></i></a>, <a class="txt" href="mesfunc5.html#E22:97"><i><font color="Green" title="E105">A17</font></i></a>, <a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2"><i><font color="Green" title="E115">A41</font></i></a>, <a class="txt" href="mesfunc5.html#E8:97_10_2_2_2_1_2"><i><font color="Green" title="E118">A44</font></i></a>, <a class="ref" href="mesfunc4.html#T3" title="MESFUNC4:th.3">MESFUNC4:3</a></i>;<br/><a NAME="E11:97_10_2_2_2_1_2"/><b>then </b><i><font color="Green" title="E119">A45</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c17">y1</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2"><i><font color="Green" title="E113">A35</font></i></a>, <a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> $1</span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c18">y2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E13:97_10_2_2_2_1_2"/><i><font color="Green" title="E120">A46</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">G2</font> &amp; ( for <font color="Olive" title="b1">j</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="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">G2</font></span>)</span> holds <br/><font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><i><font color="Green" title="E121">A47</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">G2</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">G2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:97_10_2_2_2_1_2_4"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font>
 ;<br/><a NAME="E2:97_10_2_2_2_1_2_4"/><b>then </b>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E14:97_10_2_2_2_1_2"><i><font color="Green" title="E121">A47</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:97_10_2_2_2_1_2_4"/>
<font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E16:97_10_2_2_2_1_2"/><b>then </b>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c18">y2</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:97"><i><font color="Green" title="E100">A10</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2"><i><font color="Green" title="E114">A40</font></i></a>, <a class="txt" href="mesfunc5.html#E31:97"><i><font color="Green" title="E110">A24</font></i></a>, <a class="txt" href="mesfunc5.html#E5:97_10_2_2_2_1_2"><i><font color="Green" title="E116">A42</font></i></a>, <a class="txt" href="mesfunc5.html#E14:97_10_2_2_2_1_2"><i><font color="Green" title="E121">A47</font></i></a>, <a class="ref" href="mesfunc4.html#T3" title="MESFUNC4:th.3">MESFUNC4:3</a></i>;<br/><a NAME="E17:97_10_2_2_2_1_2"/><b>then </b><i><font color="Green" title="E122">A48</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c18">y2</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2"><i><font color="Green" title="E114">A40</font></i></a>, <a class="ref" href="mesfunc5.html#D14" target="_self" title="MESFUNC5:def.14">Def14</a></i>;<br/><a NAME="E18:97_10_2_2_2_1_2"/><i><font color="Green" title="E123">A49</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_5"/><i><font color="Green" title="E124">A50</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font>
 ;<br/>

<a NAME="E2:97_10_2_2_2_1_2_5"/><b>then </b>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">G1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:97_10_2_2_2_1_2_5"/><b>then </b><i><font color="Green" title="E125">A51</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a></i>;<br/>
<i><font color="Green" title="E126">A52</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">b</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">y1</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:97"><i><font color="Green" title="E101">A11</font></i></a>, <a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E5:97_10_2_2_2_1_2_5"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">y1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_5"><i><font color="Green" title="E124">A50</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:97_10_2_2_2_1_2_5"/><b>then </b><i><font color="Green" title="E127">A53</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">y1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><i><font color="Green" title="E128">A54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_5_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2_1_2_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:97_10_2_2_2_1_2_5_2_1"/>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2_1_2_5_2_1_1"><b>suppose </b></a><a NAME="E1:97_10_2_2_2_1_2_5_2_1_1"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:97_10_2_2_2_1_2_5_2_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2_1_2_5_2_1_2"><b>suppose </b></a><a NAME="E1:97_10_2_2_2_1_2_5_2_1_2"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:97_10_2_2_2_1_2_5_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:97_10_2_2_2_1_2_5"><i><font color="Green" title="E127">A53</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:97_10_2_2_2_1_2_5_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:97_10_2_2_2_1_2_5_2_1_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E4:97_10_2_2_2_1_2_5"><i><font color="Green" title="E126">A52</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_5"><i><font color="Green" title="E124">A50</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:97_10_2_2_2_1_2_5"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">G1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_5"><i><font color="Green" title="E124">A50</font></i></a>, <a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:97_10_2_2_2_1_2_5"/><b>then </b><i><font color="Green" title="E129">A55</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">G1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:97_10_2_2_2_1_2_5"/><b>then </b><i><font color="Green" title="E130">A56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E11:97_10_2_2_2_1_2_5"/>
( <font color="Maroon" title="c14">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">G1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:97_10_2_2_2_1_2_5"><i><font color="Green" title="E129">A55</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">G1k</font> = <font color="Maroon" title="c14">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E13:97_10_2_2_2_1_2_5"/>
<font color="Maroon" title="c3">M</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E14:97_10_2_2_2_1_2_5"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">G1k</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></i>;<br/>
<b>hence </b><a NAME="E15:97_10_2_2_2_1_2_5"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2_5"><i><font color="Green" title="E125">A51</font></i></a>, <a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2_5"><i><font color="Green" title="E128">A54</font></i></a>, <a class="txt" href="mesfunc5.html#E10:97_10_2_2_2_1_2_5"><i><font color="Green" title="E130">A56</font></i></a>, <a class="ref" href="extreal2.html#T45" title="EXTREAL2:th.45">EXTREAL2:45</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:97_10_2_2_2_1_2"/><i><font color="Green" title="E124">A57</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_6"/><i><font color="Green" title="E125">A58</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font>
 ;<br/>

<a NAME="E2:97_10_2_2_2_1_2_6"/><b>then </b>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">G2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:97_10_2_2_2_1_2_6"/><b>then </b><i><font color="Green" title="E126">A59</font></i>: 
<font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a></i>;<br/>
<i><font color="Green" title="E127">A60</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">b</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">y2</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E24:97"><i><font color="Green" title="E106">A18</font></i></a>, <a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E5:97_10_2_2_2_1_2_6"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">y2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_6"><i><font color="Green" title="E125">A58</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:97_10_2_2_2_1_2_6"/><b>then </b><i><font color="Green" title="E128">A61</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">y2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><i><font color="Green" title="E129">A62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_6_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2_1_2_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:97_10_2_2_2_1_2_6_2_1"/>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2_1_2_6_2_1_1"><b>suppose </b></a><a NAME="E1:97_10_2_2_2_1_2_6_2_1_1"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:97_10_2_2_2_1_2_6_2_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2_1_2_6_2_1_2"><b>suppose </b></a><a NAME="E1:97_10_2_2_2_1_2_6_2_1_2"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:97_10_2_2_2_1_2_6_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:97_10_2_2_2_1_2_6"><i><font color="Green" title="E128">A61</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:97_10_2_2_2_1_2_6_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:97_10_2_2_2_1_2_6_2_1_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E4:97_10_2_2_2_1_2_6"><i><font color="Green" title="E127">A60</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_6"><i><font color="Green" title="E125">A58</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:97_10_2_2_2_1_2_6"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">G2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_6"><i><font color="Green" title="E125">A58</font></i></a>, <a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:97_10_2_2_2_1_2_6"/><b>then </b><i><font color="Green" title="E130">A63</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">G2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:97_10_2_2_2_1_2_6"/><b>then </b><i><font color="Green" title="E131">A64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E11:97_10_2_2_2_1_2_6"/>
( <font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">G2</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:97_10_2_2_2_1_2_6"><i><font color="Green" title="E130">A63</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">G2k</font> = <font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E13:97_10_2_2_2_1_2_6"/>
<font color="Maroon" title="c3">M</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E14:97_10_2_2_2_1_2_6"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">G2k</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</a></i>;<br/>
<b>hence </b><a NAME="E15:97_10_2_2_2_1_2_6"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2_6"><i><font color="Green" title="E126">A59</font></i></a>, <a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2_6"><i><font color="Green" title="E129">A62</font></i></a>, <a class="txt" href="mesfunc5.html#E10:97_10_2_2_2_1_2_6"><i><font color="Green" title="E131">A64</font></i></a>, <a class="ref" href="extreal2.html#T45" title="EXTREAL2:th.45">EXTREAL2:45</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green" title="E125">A65</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">y2</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:97"><i><font color="Green" title="E101">A11</font></i></a>, <a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a>, <a class="txt" href="mesfunc5.html#E24:97"><i><font color="Green" title="E106">A18</font></i></a>, <a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E21:97_10_2_2_2_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c18">y2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_8"><b>proof </b></a><div class="add">

<a NAME="E1:97_10_2_2_2_1_2_8"/><i><font color="Green" title="E126">A66</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font> holds <br/> not <font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_8_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_8_1"/><i><font color="Green" title="E127">A67</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:97_10_2_2_2_1_2_8_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E18:97_10_2_2_2_1_2"><i><font color="Green" title="E123">A49</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_8_1"><i><font color="Green" title="E127">A67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:97_10_2_2_2_1_2_8_1"/>
not <font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:97_10_2_2_2_1_2_8"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_8_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:97_10_2_2_2_1_2_8_2"/>
(  not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font> or  not <font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_8"><i><font color="Green" title="E126">A66</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:97_10_2_2_2_1_2_8_2"/>
not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:97_10_2_2_2_1_2_8"/><b>then </b><i><font color="Green" title="E127">A68</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:97_10_2_2_2_1_2_8"/><i><font color="Green" title="E128">A69</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font> holds <br/> not <font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_8_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_8_3"/><i><font color="Green" title="E129">A70</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:97_10_2_2_2_1_2_8_3"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:97_10_2_2_2_1_2"><i><font color="Green" title="E124">A57</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_8_3"><i><font color="Green" title="E129">A70</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:97_10_2_2_2_1_2_8_3"/>
not <font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:97_10_2_2_2_1_2_8"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">y2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_8_4"><b>proof </b></a><div class="add">

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

<a NAME="E1:97_10_2_2_2_1_2_8_4"/>
(  not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font> or  not <font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:97_10_2_2_2_1_2_8"><i><font color="Green" title="E128">A69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:97_10_2_2_2_1_2_8_4"/>
not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">y2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:97_10_2_2_2_1_2_8"/><b>then </b><i><font color="Green" title="E129">A71</font></i>: 
<font color="Maroon" title="c18">y2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<i><font color="Green" title="E130">A72</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">y</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E11:97"><i><font color="Green" title="E98">A8</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">G2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="mesfunc5.html#E15:97"><i><font color="Green" title="E101">A11</font></i></a>, <a class="txt" href="mesfunc5.html#E24:97"><i><font color="Green" title="E106">A18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E8:97_10_2_2_2_1_2_8"/><b>then </b><i><font color="Green" title="E131">A73</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">y</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">y2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">y2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:97_10_2_2_2_1_2_8"><i><font color="Green" title="E129">A71</font></i></a>, <a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2_8"><i><font color="Green" title="E127">A68</font></i></a></i>;<br/>
<a NAME="E9:97_10_2_2_2_1_2_8"/>
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>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">y</font> holds <br/><font color="Maroon" title="c12">y</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_8_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_8_6"/><i><font color="Green" title="E132">A74</font></i>: 
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">y</font>
 ;<br/>

<a NAME="E2:97_10_2_2_2_1_2_8_6"/><b>then </b><i><font color="Green" title="E133">A75</font></i>: 
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:97"><i><font color="Green" title="E97">A7</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:97_10_2_2_2_1_2_8_6"/><b>then </b><i><font color="Green" title="E134">A76</font></i>: 
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:97_10_2_2_2_1_2_8_6"/>
( <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">y2</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2_8"><i><font color="Green" title="E130">A72</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E132">A74</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:97_10_2_2_2_1_2_8_6"/><b>then </b><i><font color="Green" title="E135">A77</font></i>: 
( <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">G1</font></span>)</span> &amp; <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">G2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a>, <a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:97_10_2_2_2_1_2_8_6"/><b>then </b><i><font color="Green" title="E136">A78</font></i>: 
( <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">G1</font> &amp; <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">G2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:97_10_2_2_2_1_2_8_6"/><i><font color="Green" title="E137">A79</font></i>: 
<font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E134">A76</font></i></a>, <a class="txt" href="mesfunc5.html#E26:97"><i><font color="Green" title="E108">A20</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_8_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:97_10_2_2_2_1_2_8_6_1"/><i><font color="Green" title="E138">A80</font></i>: 
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font>
 ;<br/><a NAME="E2:97_10_2_2_2_1_2_8_6_1"/>
<font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E134">A76</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:97_10_2_2_2_1_2_8_6_1"/><b>then </b>
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2_8_6_1"><i><font color="Green" title="E138">A80</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E4:97_10_2_2_2_1_2_8_6_1"/><b>then </b>
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:97"><i><font color="Green" title="E96">A6</font></i></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></i>;<br/><a NAME="E5:97_10_2_2_2_1_2_8_6_1"/><b>then </b>
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E6:97_10_2_2_2_1_2_8_6_1"/>
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:97_10_2_2_2_1_2_8_6"/><b>then </b>
<font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E138">A81</font></i>: <font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E134">A76</font></i></a>, <a class="txt" href="mesfunc5.html#E17:97"><i><font color="Green" title="E103">A13</font></i></a>, <a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E137">A79</font></i></a></i>
;<br/>
<a NAME="E11:97_10_2_2_2_1_2_8_6"/>
( <font color="Maroon" title="c14">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">G1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E136">A78</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">G1n</font> = <font color="Maroon" title="c14">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E13:97_10_2_2_2_1_2_8_6"/>
( <font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">G2</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E136">A78</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">G2n</font> = <font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<div><i><font color="Green" title="E139">A82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_10_2_2_2_1_2_8_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_10_2_2_2_1_2_8_6_3"/>
<font color="Maroon" title="c14">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c22">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:97_10_2_2_2_1_2_8_6_3"/><i><font color="Green" title="E140">A83</font></i>: 
( <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> &amp; <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:97_10_2_2_2_1_2_8_6_3"/>
( <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2_8_6_3"><i><font color="Green" title="E140">A83</font></i></a>, <a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E134">A76</font></i></a>, <a class="txt" href="mesfunc5.html#E17:97"><i><font color="Green" title="E103">A13</font></i></a>, <a class="txt" href="mesfunc5.html#E26:97"><i><font color="Green" title="E108">A20</font></i></a></i>;<br/><a NAME="E5:97_10_2_2_2_1_2_8_6_3"/><b>then </b>
( <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c22">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:97_10_2_2_2_1_2_8_6_3"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green" title="E140">A84</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c10">G</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> = 
<font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E134">A76</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">G1n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">G2n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E138">A81</font></i></a>, <a class="txt" href="mesfunc5.html#E15:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E139">A82</font></i></a>, <a class="ref" href="measure1.html#T61" title="MEASURE1:th.61">MEASURE1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E136">A78</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E136">A78</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E17:97_10_2_2_2_1_2_8_6"/>
<font color="Maroon" title="c3">M</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E18:97_10_2_2_2_1_2_8_6"/><b>then </b>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">G1n</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">G2n</font> )
 
<b>by </b><i><a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>;<br/>
<a NAME="E19:97_10_2_2_2_1_2_8_6"/><b>then </b>
( ( 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> or 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> ) &amp; ( 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> or 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> ) )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E136">A78</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E20:97_10_2_2_2_1_2_8_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c10">G</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E16:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E140">A84</font></i></a>, <a class="ref" href="extreal1.html#T29" title="EXTREAL1:th.29">EXTREAL1:29</a></i>;<br/>
<a NAME="E21:97_10_2_2_2_1_2_8_6"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c14">G1</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E133">A75</font></i></a>, <a class="txt" href="mesfunc5.html#E10:97"><i><font color="Green" title="E97">A7</font></i></a></i>;<br/>
<a NAME="E22:97_10_2_2_2_1_2_8_6"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">G2</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E135">A77</font></i></a>, <a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green" title="E117">A43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:97_10_2_2_2_1_2_8_6"/>
<font color="Maroon" title="c12">y</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">y2</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:97_10_2_2_2_1_2_8_6"><i><font color="Green" title="E135">A77</font></i></a>, <a class="txt" href="mesfunc5.html#E13:97_10_2_2_2_1_2"><i><font color="Green" title="E120">A46</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:97_10_2_2_2_1_2_8"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c18">y2</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:97_10_2_2_2_1_2_8"><i><font color="Green" title="E131">A73</font></i></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E22:97_10_2_2_2_1_2"/>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:97_10_2"><i><font color="Green" title="E112">A30</font></i></a>, <a class="txt" href="mesfunc5.html#E11:97_10_2_2_2_1_2"><i><font color="Green" title="E119">A45</font></i></a>, <a class="txt" href="mesfunc5.html#E17:97_10_2_2_2_1_2"><i><font color="Green" title="E122">A48</font></i></a>, <a class="txt" href="mesfunc5.html#E18:97_10_2_2_2_1_2"><i><font color="Green" title="E123">A49</font></i></a>, <a class="txt" href="mesfunc5.html#E19:97_10_2_2_2_1_2"><i><font color="Green" title="E124">A57</font></i></a>, <a class="txt" href="mesfunc5.html#E20:97_10_2_2_2_1_2"><i><font color="Green" title="E125">A65</font></i></a>, <a class="ref" href="mesfunc4.html#T1" title="MESFUNC4:th.1">MESFUNC4:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
