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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">B</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>and </b><br/><a NAME="E3:97"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/>

<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">g1</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">g2</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font>;<br/>
<a NAME="E4:97"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="txt" href="mesfunc5.html#E12:97"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E5:97"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:97"/><b>then </b><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E8:97"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">G</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">b</font> holds <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:97"><i><font color="Green">E94</font></i></a>, <a class="ref" href="mesfunc3.html#T14">MESFUNC3:14</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E10:97"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<i><font color="Green">E43</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>
;<br/>
<a NAME="E12:97"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="txt" href="mesfunc5.html#E12:97"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E13:97"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a>, <a class="txt" href="mesfunc5.html#E12:97"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">G1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E15:97"/><i><font color="Green">E94</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> holds <br/><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<i><font color="Green">E95</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E17:97"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">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">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:97_3"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:97_3"/>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G1</font> as   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E22:97"><i><font color="Green">E102</font></i></a>, </i>;<br/>
<i><font color="Green">E99</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E20:97"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_5"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E2:97_5"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:97_5"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E26:97"><i><font color="Green">E106</font></i></a>, <a class="ref" href="funct_1.html#T70">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E25:97"><i><font color="Green">E104</font></i></a>, , <a class="ref" href="funct_1.html#T70">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E25:97"><i><font color="Green">E104</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E22:97"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">G1</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E22:97"><i><font color="Green">E102</font></i></a>, <a class="txt" href="mesfunc5.html#E24:97"><i><font color="Green">E103</font></i></a>, </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>;<br/>
<b>consider </b><font color="Maroon">G2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E24:97"/><i><font color="Green">E103</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> holds <br/><font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<i><font color="Green">E104</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E26:97"><i><font color="Green">E106</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E26:97"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">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">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_7"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:97_7"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:97_7"/>
<font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E26:97"><i><font color="Green">E106</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G2</font> = <font color="Maroon">G2</font> as   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E28:97"><i><font color="Green">E107</font></i></a>, </i>;<br/>
<i><font color="Green">E107</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E29:97"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_9"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<a NAME="E2:97_9"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:97_9"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_9"><i><font color="Green">E109</font></i></a>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="funct_1.html#T70">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E31:97"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">G2</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E28:97"><i><font color="Green">E107</font></i></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">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">E122</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_1"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E3:97_10_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">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">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E5:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E6:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E7:97_10_1"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E8:97_10_1"/><i><font color="Green">E125</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E9:97_10_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">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">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E11:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E12:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E13:97_10_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E14:97_10_1"/><b>then </b>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E15:97_10_1"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2"><b>suppose </b></a><a NAME="E1:97_10_2"/><i><font color="Green">E126</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_2"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">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">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_10_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:97_10_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:97_10_2_1"/>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E3:97_10_2"/><b>then </b>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:97"><i><font color="Green">E94</font></i></a>, <a class="txt" href="mesfunc5.html#E16:97"><i><font color="Green">E95</font></i></a>, , <a class="txt" href="mesfunc5.html#E17:97"><i><font color="Green">E98</font></i></a>, , <a class="ref" href="mesfunc4.html#T3">MESFUNC4:3</a></i>;<br/><a NAME="E4:97_10_2"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</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">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">E128</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_2_2_1"/><i><font color="Green">E129</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><i><font color="Green">E130</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:97_10_2_2_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="funct_1.html#T70">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E6:97_10_2_2_1"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_10_2_2_2"><b>suppose </b></a><a NAME="E1:97_10_2_2_2"/><i><font color="Green">E132</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">E133</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_2_2_2_1_1"/><i><font color="Green">E134</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><i><font color="Green">E135</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="funct_1.html#T68">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:97_10_2_2_2_1_1_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="funct_1.html#T70">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">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E6:97_10_2_2_2_1_1"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E137</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:97_10_2_2_2_1_2"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i>, </i>;<br/><a NAME="E3:97_10_2_2_2_1_2"/><b>then </b><i><font color="Green">E138</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/><a NAME="E4:97_10_2_2_2_1_2"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i>, </i>;<br/><a NAME="E5:97_10_2_2_2_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">y1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E7:97_10_2_2_2_1_2"/><i><font color="Green">E140</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> holds <br/><font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><i><font color="Green">E141</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">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">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:97_10_2_2_2_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font>
 ;<br/><a NAME="E2:97_10_2_2_2_1_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:97_10_2_2_2_1_2_2"/>
<font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</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">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E19:97"><i><font color="Green">E99</font></i></a>, , , <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, , <a class="ref" href="mesfunc4.html#T3">MESFUNC4:3</a></i>;<br/><a NAME="E11:97_10_2_2_2_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y1</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">y2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E13:97_10_2_2_2_1_2"/><i><font color="Green">E143</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font></span>)</span> holds <br/><font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><i><font color="Green">E184</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">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">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:97_10_2_2_2_1_2_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font>
 ;<br/><a NAME="E2:97_10_2_2_2_1_2_4"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E31:97"><i><font color="Green">E121</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:97_10_2_2_2_1_2_4"/>
<font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</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">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y2</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="txt" href="mesfunc5.html#E31:97"><i><font color="Green">E121</font></i></a>, <a class="ref" href="mesfunc4.html#T3">MESFUNC4:3</a></i>;<br/><a NAME="E17:97_10_2_2_2_1_2"/><b>then </b><i><font color="Green">E185</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E18:97_10_2_2_2_1_2"/><i><font color="Green">E186</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">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">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_5"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font>
 ;<br/>

<a NAME="E2:97_10_2_2_2_1_2_5"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#D3">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">E188</font></i>: 
<font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a></i>;<br/>
<i><font color="Green">E189</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">b</font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E5:97_10_2_2_2_1_2_5"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">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">E190</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><i><font color="Green">E191</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">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&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">k</font> <a href="hidden.html#R1">=</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">&lt;=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><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">k</font> <a href="hidden.html#NR2" target="_self">&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">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:97_10_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="real_1.html#D5">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">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">&lt;=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E1:97_10_2"><i><font color="Green">E126</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">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#D3">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">E192</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">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">E193</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E11:97_10_2_2_2_1_2_5"/>
( <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G1k</font> = <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E13:97_10_2_2_2_1_2_5"/>
<font color="Maroon">M</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11">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">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">G1k</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">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">&lt;=</a> <font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:97_10_2_2_2_1_2"/><i><font color="Green">E194</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">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">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_6"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font>
 ;<br/>

<a NAME="E2:97_10_2_2_2_1_2_6"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_1.html#D3">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">E196</font></i>: 
<font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/>
<i><font color="Green">E197</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">b</font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E26:97"><i><font color="Green">E106</font></i></a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E5:97_10_2_2_2_1_2_6"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">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">E198</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><i><font color="Green">E199</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">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&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">k</font> <a href="hidden.html#R1">=</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">&lt;=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><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">k</font> <a href="hidden.html#NR2" target="_self">&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">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">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">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">&lt;=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E4:97_10_2"><i><font color="Green">E127</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">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_1.html#D3">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">E200</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">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">E201</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E11:97_10_2_2_2_1_2_6"/>
( <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G2k</font> = <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E13:97_10_2_2_2_1_2_6"/>
<font color="Maroon">M</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11">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">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">G2k</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">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">&lt;=</a> <font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2"><i><font color="Green">E126</font></i></a>, , , <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E202</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="txt" href="mesfunc5.html#E26:97"><i><font color="Green">E106</font></i></a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E21:97_10_2_2_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">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">E203</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font> holds <br/> not <font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_8_1"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:97_10_2_2_2_1_2_8_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E4:97_10_2"><i><font color="Green">E127</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:97_10_2_2_2_1_2_8_1"/>
not <font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">x</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:97_10_2_2_2_1_2_8_2"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font> or  not <font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:97_10_2"><i><font color="Green">E126</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:97_10_2_2_2_1_2_8_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">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">E205</font></i>: 
<font color="Maroon">y1</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:97_10_2_2_2_1_2_8"/><i><font color="Green">E206</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font> holds <br/> not <font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_8_3"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:97_10_2_2_2_1_2_8_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:97_10_2_2_2_1_2_8_3"/>
not <font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">x</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y2</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:97_10_2_2_2_1_2_8_4"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font> or  not <font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:97_10_2_2_2_1_2_8_4"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y2</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">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">E208</font></i>: 
<font color="Maroon">y2</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<i><font color="Green">E209</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E17:97"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , <a class="txt" href="mesfunc5.html#E26:97"><i><font color="Green">E106</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_1.html#D3">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">E210</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y1</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">y2</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y1</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">y2</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E4:97_10_2"><i><font color="Green">E127</font></i></a></i>;<br/>
<a NAME="E9:97_10_2_2_2_1_2_8"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:97_10_2_2_2_1_2_8_6"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:97_10_2_2_2_1_2_8_6"/><b>then </b><i><font color="Green">E305</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">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">E306</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:97_10_2_2_2_1_2_8_6"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y2</font> )
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="xboole_0.html#D3">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">E307</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="ref" href="finseq_1.html#D3">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">E308</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:97_10_2_2_2_1_2_8_6"/><i><font color="Green">E309</font></i>: 
<font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </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">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:97_10_2_2_2_1_2_8_6_1"/><i><font color="Green">E310</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:97_10_2_2_2_1_2_8_6_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">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">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2"><i><font color="Green">E138</font></i></a>, <a class="ref" href="tarski.html#D4">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">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">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">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">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">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">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">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E311</font></i>: <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E24:97"><i><font color="Green">E103</font></i></a>, <a class="txt" href="mesfunc5.html#E1:97_10_2_2_2_1_2"><i><font color="Green">E137</font></i></a></i>
;<br/>
<a NAME="E11:97_10_2_2_2_1_2_8_6"/>
( <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G1n</font> = <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E13:97_10_2_2_2_1_2_8_6"/>
( <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G2n</font> = <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<div><i><font color="Green">E312</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">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:97_10_2_2_2_1_2_8_6_3"/><i><font color="Green">E313</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:97_10_2_2_2_1_2_8_6_3"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E7:97_10_2_2_2_1_2"><i><font color="Green">E140</font></i></a>, , <a class="txt" href="mesfunc5.html#E24:97"><i><font color="Green">E103</font></i></a>, </i>;<br/><a NAME="E5:97_10_2_2_2_1_2_8_6_3"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">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="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green">E314</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">G1n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">G2n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:97_10_2_2_2_1_2"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E5:97_10_2_2_2_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="measure1.html#T61">MEASURE1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E17:97_10_2_2_2_1_2_8_6"/>
<font color="Maroon">M</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11">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">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">G1n</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">G2n</font> )
 
<b>by </b><i><a class="ref" href="measure1.html#D4">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">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> ) &amp; ( 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> ) )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">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">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</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">E140</font></i></a>, <a class="ref" href="extreal1.html#T29">EXTREAL1:29</a></i>;<br/>
<a NAME="E21:97_10_2_2_2_1_2_8_6"/><b>then </b>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E22:97_10_2_2_2_1_2_8_6"/><b>then </b>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E23:97_10_2_2_2_1_2_8_6"/>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</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">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">y2</font>
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D3">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" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, , , , , <a class="ref" href="mesfunc4.html#T1">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>
