<?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><a NAME="E1:114"/><i><font color="Green">E35</font></i>: 
(  ex <font color="Olive">E</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">E</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">E</font> ) &amp; <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">E</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:114"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:114"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font>
 ;<br/>
<b>set </b><font color="Maroon">C</font> = <font color="Maroon">E</font> <a href="measure1.html#K3">/\</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/>
<a NAME="E5:114"/><i><font color="Green">E39</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> <font color="Maroon">E</font> <a href="measure1.html#K3">/\</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="E6:114"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">E</font> <a href="measure1.html#K3">/\</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="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:114"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">E</font> <a href="measure1.html#K3">/\</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> <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">E</font> <a href="measure1.html#K3">/\</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_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E8:114"/><b>then </b><i><font color="Green">E42</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="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">E</font> <a href="measure1.html#K3">/\</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>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E9:114"/>
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/><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> <a href="hidden.html#R1">=</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">E</font> <a href="measure1.html#K3">/\</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="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_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="E1:114_1"/><i><font color="Green">E43</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="E2:114_1"/><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="E3:114_1"/>
<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> <span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K3">/\</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="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:114"/><b>then </b><i><font color="Green">E44</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="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K3">/\</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="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E11:114"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> <a href="measure1.html#K3">/\</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="mesfunc5.html#T26" target="_self">Th26</a>, , <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="mesfunc1.html#T34">MESFUNC1:34</a></i>;<br/>
<a NAME="E12:114"/><b>then </b><i><font color="Green">E47</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="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> <a href="measure1.html#K3">/\</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="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, , </i>;<br/>
<a NAME="E13:114"/><i><font color="Green">E94</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> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">F0</font> being    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">K0</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E15:114"/><i><font color="Green">E95</font></i>: 
( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> ) ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a> ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/>( <font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">K0</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) &amp; <font color="Maroon">K0</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K0</font> )
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E21:114"><i><font color="Green">E101</font></i></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="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <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>consider </b><font color="Maroon">FAB</font> being    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E17:114"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1">SEQFUNC:sch 1</a>();<br/></i>
<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="numbers.html#K6">ExtREAL</a>  =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">KAB</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E19:114"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">KAB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:114_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:114_2"/>
<font color="Maroon">KAB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:114"/><i><font color="Green">E101</font></i>: 
( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</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> ) ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  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/><span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a> ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  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/><span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  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/>( <font color="Maroon">FAB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</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> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_3"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:114_3_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:114_3_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E3:114_3_1"/>
<font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:114_3_1"/>
<font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E10:114"><i><font color="Green">E44</font></i></a></i>;<br/><b>thus </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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="mesfunc5.html#T30" target="_self">Th30</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></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"><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/>
<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> <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/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:114_3_2"/>
<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="E2:114_3_2"/><b>then </b><i><font color="Green">E107</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">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:114_3_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:114_3_2"/><b>then </b>
<font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>;<br/><b>hence </b><a NAME="E5:114_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:114_3_3"/>
<font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i/>;<br/><a NAME="E3:114_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> <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/>;<br/><b>hence </b><a NAME="E4:114_3_3"/>
<font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:114_3_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_3_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:114_3_4_1"/><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>
 ;<br/><a NAME="E2:114_3_4_1"/><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> <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="E3:114_3_4_1"/><b>then </b><i><font color="Green">E121</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">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:114_3_4_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:114_3_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:114_3_4"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  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/><span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 ;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:114_3_5"/><i><font color="Green">E122</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="E2:114_3_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> <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="E3:114_3_5"/><b>then </b><i><font color="Green">E123</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">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_3_5_1"><b>now </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>thus </b><span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:114_3_5"><i><font color="Green">E124</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>
;<br/></div><b>end;</b></div><a NAME="E5:114_3_5"/><b>then </b>
<font color="Maroon">FAB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E6:114_3_5"/>
<font color="Maroon">FAB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i>, </i>;<br/><b>thus </b> <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<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#T68">FUNCT_1:68</a></i>
;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E22:114"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">KAB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">KAB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_4"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font>, <font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:114_4"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</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> &amp; <font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:114_4"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">KAB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:114_4"/><i><font color="Green">E128</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"><span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_4_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="E1:114_4_1"/>
<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">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_4_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></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"><span class="p4">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:114_4_1"/><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">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:114_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:114_4"/><b>then </b><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"><span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E13:114"><i><font color="Green">E94</font></i></a></i>;<br/>
<a NAME="E7:114_4"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E3:114_4"><i><font color="Green">E126</font></i></a>, </i>;<br/>
<a NAME="E8:114_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, </i>;<br/>
<b>hence </b><a NAME="E9:114_4"/>
<font color="Maroon">KAB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">KAB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E1:114_4"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_4"><i><font color="Green">E127</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:114"/><b>then </b>
<font color="Maroon">KAB</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E24:114"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="mesfunc5.html#K6" 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="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">KAB</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E21:114"><i><font color="Green">E101</font></i></a>, , , <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, , <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a></i>;<br/>
<b>set </b><font color="Maroon">DA</font> = <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">DB</font> = <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font>;<br/>
<a NAME="E25:114"/>
( <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> &amp; <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E26:114"/><b>then </b><i><font color="Green">E131</font></i>: 
( <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="mesfunc1.html#T34">MESFUNC1:34</a></i>;<br/>
<a NAME="E27:114"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</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> &amp; <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</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> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E28:114"/><b>then </b>
( <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E29:114"/><b>then </b><i><font color="Green">E133</font></i>: 
( <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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font> &amp; <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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E30:114"/><b>then </b><i><font color="Green">E134</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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E31:114"/>
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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> holds <br/><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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font></span>)</span></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="114_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="E1:114_5"/><i><font color="Green">E135</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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_5"/><b>then </b>
<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> <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="E3:114_5"/>
<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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</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="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:114"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</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="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E33:114"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E34:114"/>
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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> holds <br/><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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font></span>)</span></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="114_6"><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="E1:114_6"/><i><font color="Green">E137</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">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_6"/><b>then </b>
<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> <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="E3:114_6"/>
<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">E</font> <a href="measure1.html#K3">/\</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="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:114"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="measure1.html#K3">/\</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="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E36:114"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E37:114"/><i><font color="Green">E139</font></i>: 
( <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> &amp; <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/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">FA</font> being    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E39:114"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1">SEQFUNC:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font>;<br/>
<b>consider </b><font color="Maroon">FB</font> being    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E41:114"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1">SEQFUNC:sch 1</a>();<br/></i>
<a NAME="E42:114"/><i><font color="Green">E142</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_7"><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>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:114_7"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:114_7"/><i><font color="Green">E184</font></i>: 
( <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E131</font></i></a>, <a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green">E132</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:114_7"/>
( <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:114"><i><font color="Green">E44</font></i></a>, <a class="txt" href="mesfunc5.html#E30:114"><i><font color="Green">E134</font></i></a></i>;<br/>

<a NAME="E5:114_7"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</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/>
<b>hence </b><a NAME="E6:114_7"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</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> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:114"/><i><font color="Green">E185</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_8"><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>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:114_8"/>
<font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:114_8"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:114_8"/>
( <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E131</font></i></a>, <a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green">E132</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:114"/><i><font color="Green">E186</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  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/><span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_9"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:114_9"/><i><font color="Green">E188</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> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E3:114_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> <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="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:114_9"/><b>then </b><i><font color="Green">E189</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">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font>, <font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:114_9"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E7:114_9"/><b>then </b><i><font color="Green">E190</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</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> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E133</font></i></a></i>;<br/>
<a NAME="E8:114_9"/>
( <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</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"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></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> &amp; <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</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"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></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> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E9:114_9"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E37:114"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E10:114_9"/>
<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E33:114"><i><font color="Green">E136</font></i></a>, <a class="txt" href="mesfunc5.html#E36:114"><i><font color="Green">E138</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:114"/><i><font color="Green">E191</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  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/><span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_10"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:114_10"/><i><font color="Green">E193</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> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<a NAME="E3:114_10"/><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> <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="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:114_10"/><b>then </b><i><font color="Green">E194</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">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font>, <font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:114_10"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green">E132</font></i></a></i>;<br/>
<a NAME="E7:114_10"/><b>then </b><i><font color="Green">E195</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</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> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E133</font></i></a></i>;<br/>
<a NAME="E8:114_10"/>
( <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</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"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></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> &amp; <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</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"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></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> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green">E132</font></i></a></i>;<br/>
<a NAME="E9:114_10"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E36:114"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green">E140</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E10:114_10"/>
<span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E37:114"><i><font color="Green">E139</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:114"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  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/>( <font color="Maroon">FA</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> <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="114_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:114_11"/><i><font color="Green">E197</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> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E2:114_11"/><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> <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="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:114_11"/><b>then </b><i><font color="Green">E198</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">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_11_1"><b>now </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/><i><font color="Green">E199</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E131</font></i></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="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E133</font></i></a></i>
;<br/><a NAME="E2:114_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <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">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E3:114_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></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>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E131</font></i></a></i>;<br/><a NAME="E4:114_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E36:114"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green">E140</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E5:114_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <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">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:114_11"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">FA</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E6:114_11"/>
( <font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <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="txt" href="mesfunc5.html#E37:114"><i><font color="Green">E139</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:114_11"/>
( <font color="Maroon">FA</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <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#E39:114"><i><font color="Green">E140</font></i></a>, <a class="txt" href="mesfunc5.html#E36:114"><i><font color="Green">E138</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E47:114"/><i><font color="Green">E201</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  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/>( <font color="Maroon">FB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> <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="114_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:114_12"/><i><font color="Green">E202</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> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<a NAME="E2:114_12"/><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> <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="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:114_12"/><b>then </b><i><font color="Green">E203</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">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_12_1"><b>now </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/><i><font color="Green">E204</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">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green">E132</font></i></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="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E133</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></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>
<b>by </b><i><a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green">E132</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E37:114"><i><font color="Green">E139</font></i></a>, <a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green">E141</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E5:114_12"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">FB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E6:114_12"/>
( <font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <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="txt" href="mesfunc5.html#E39:114"><i><font color="Green">E140</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:114_12"/>
( <font color="Maroon">FB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <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#E41:114"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E37:114"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<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>  =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">KA</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E49:114"/><i><font color="Green">E206</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green">E207</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:114_13"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:114_13"/>
<font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i><a class="txt" href="mesfunc5.html#E37:114"><i><font color="Green">E139</font></i></a></i>;<br/></div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>6</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>  =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">KB</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E52:114"/><i><font color="Green">E208</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green">E209</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:114_14"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:114_14"/>
<font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green">E141</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E54:114"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>( <font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> &amp; <font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_15"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:114_15"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</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> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E133</font></i></a></i>;<br/>
<a NAME="E3:114_15"/><i><font color="Green">E305</font></i>: 
( <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:114"><i><font color="Green">E134</font></i></a></i>;<br/>
<a NAME="E4:114_15"/><i><font color="Green">E306</font></i>: 
( <font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green">E140</font></i></a>, <a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green">E142</font></i></a></i>;<br/>
<a NAME="E5:114_15"/><i><font color="Green">E307</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"><span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_15_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="E1:114_15_1"/>
<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">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_15_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></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"><span class="p4">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:114_15_1"/><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">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:114_15_1"/>
<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:114_15"/><b>then </b><i><font color="Green">E308</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, , <a class="txt" href="mesfunc5.html#E13:114"><i><font color="Green">E94</font></i></a></i>;<br/>
<a NAME="E7:114_15"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, , <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, </i>;<br/>
<a NAME="E8:114_15"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E9:114_15"/>
<font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green">E140</font></i></a>, <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a></i>;<br/>

<a NAME="E10:114_15"/><i><font color="Green">E309</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"><span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_15_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/>

<b>assume </b><a NAME="E1:114_15_2"/>
<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">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_15_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></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"><span class="p4">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:114_15_2"/><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">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:114_15_2"/>
<span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E33:114"><i><font color="Green">E136</font></i></a>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:114_15"/><b>then </b><i><font color="Green">E310</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, , <a class="txt" href="mesfunc5.html#E13:114"><i><font color="Green">E94</font></i></a></i>;<br/>
<a NAME="E12:114_15"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, , , </i>;<br/>
<a NAME="E13:114_15"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E14:114_15"/>
<font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green">E142</font></i></a>, <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:114"/><b>then </b><i><font color="Green">E311</font></i>: 
( ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> ) )
 
;<br/>
<a NAME="E56:114"/><b>then </b>
( <font color="Maroon">KA</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp; <font color="Maroon">KB</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E57:114"/><b>then </b><i><font color="Green">E312</font></i>: 
(  <a href="mesfunc5.html#K6" 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="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">KA</font> &amp;  <a href="mesfunc5.html#K6" 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="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">KB</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E21:114"><i><font color="Green">E101</font></i></a>, , , , <a class="txt" href="mesfunc5.html#E24:114"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E133</font></i></a>, <a class="txt" href="mesfunc5.html#E30:114"><i><font color="Green">E134</font></i></a>, , , <a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green">E140</font></i></a>, <a class="txt" href="mesfunc5.html#E33:114"><i><font color="Green">E136</font></i></a>, <a class="txt" href="mesfunc5.html#E36:114"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green">E142</font></i></a></i>;<br/>
<a NAME="E58:114"/><i><font color="Green">E313</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">KAB</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">KA</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">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_16"><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>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:114_16"/><i><font color="Green">E314</font></i>: 
( <font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:114_16"/>
( <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E131</font></i></a>, <a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green">E132</font></i></a></i>;<br/>
<a NAME="E4:114_16"/><b>then </b><i><font color="Green">E315</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, , </i>;<br/>
<i><font color="Green">E316</font></i>: <font color="Maroon">KAB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> = 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FAB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F0</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <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="mesfunc5.html#T30" target="_self">Th30</a></i>
;<br/>
<a NAME="E6:114_16"/>
<font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green">E140</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:114_16"/>
<font color="Maroon">KAB</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">KA</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">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green">E142</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E59:114"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">KA</font> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</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">KB</font> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:114_17_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">KA</font>
 ;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:114_17_1"/><i><font color="Green">E317</font></i>: 
( <font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E133</font></i></a>, <a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green">E140</font></i></a></i>;<br/><a NAME="E4:114_17_1"/>
<font color="Maroon">FA</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:114"><i><font color="Green">E134</font></i></a></i>;<br/><b>hence </b><a NAME="E5:114_17_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">KA</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:114_17"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">KB</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E4:114_17"/><i><font color="Green">E318</font></i>: 
( <font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E133</font></i></a>, <a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green">E142</font></i></a></i>;<br/>
<a NAME="E5:114_17"/>
<font color="Maroon">FB</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:114"><i><font color="Green">E134</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:114_17"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">KB</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E60:114"/><b>then </b>
( <font color="Maroon">KA</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp; <font color="Maroon">KB</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E61:114"/>
 <a href="mesfunc5.html#K6" 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#K6" 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#K6" 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#T32" target="_self">Th32</a>, <a class="ref" href="mesfunc5.html#T17" target="_self">Th17</a>, , </i>;<br/>


</div>
