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

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









<b>assume </b><a NAME="E1:114"/><i><font color="Green" title="E108">A1</font></i>: 
(  ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Olive" title="b1">E</font> ) &amp; <font color="Maroon" title="c4">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">B</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:114"/><i><font color="Green" title="E109">A2</font></i>: 
<font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E4:114"/><i><font color="Green" title="E110">A3</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c7">E</font>
 ;<br/>
<b>set </b><font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>;<br/>
<a NAME="E5:114"/><i><font color="Green" title="E111">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green" title="E109">A2</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:114"/><i><font color="Green" title="E112">A5</font></i>: 
<font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green" title="E109">A2</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:114"/><b>then </b><i><font color="Green" title="E113">A6</font></i>: 
<font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E8:114"/><b>then </b><i><font color="Green" title="E114">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:114"><i><font color="Green" title="E111">A4</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E9:114"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:114_1"/><i><font color="Green" title="E115">A8</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E3:114_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:114"><i><font color="Green" title="E114">A7</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_1"><i><font color="Green" title="E115">A8</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:114"/><b>then </b><i><font color="Green" title="E115">A9</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:114"><i><font color="Green" title="E114">A7</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E11:114"/>
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:114"><i><font color="Green" title="E112">A5</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green" title="E109">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114"><i><font color="Green" title="E110">A3</font></i></a>, <a class="ref" href="mesfunc1.html#T34" title="MESFUNC1:th.34">MESFUNC1:34</a></i>;<br/>
<a NAME="E12:114"/><b>then </b><i><font color="Green" title="E116">A10</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:114"><i><font color="Green" title="E115">A9</font></i></a>, <a class="txt" href="mesfunc5.html#E7:114"><i><font color="Green" title="E113">A6</font></i></a>, <a class="ref" href="mesfunc5.html#T48" target="_self" title="MESFUNC5:th.48">Th48</a></i>;<br/>
<a NAME="E13:114"/><i><font color="Green" title="E117">A11</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:114"><i><font color="Green" title="E108">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">F0</font> being    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> , <font color="Maroon" title="c10">K0</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E15:114"/><i><font color="Green" title="E118">A12</font></i>: 
( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> ) ) &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/>( <font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> ) ) &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c10">K0</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c10">K0</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c10">K0</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114"><i><font color="Green" title="E108">A1</font></i></a>, <a class="ref" href="mesfunc5.html#D15" target="_self" title="MESFUNC5:def.15">Def15</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c11">FAB</font> being    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E17:114"/><i><font color="Green" title="E119">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1" title="SEQFUNC:sch.1">SEQFUNC:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  =  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c12">KAB</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E19:114"/><i><font color="Green" title="E120">A14</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green" title="E121">A15</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" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:114_2"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:114_2"/>
<font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c13">n</font>)
 <b>by </b><i><a class="txt" href="mesfunc5.html#E19:114"><i><font color="Green" title="E120">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:114"/><i><font color="Green" title="E122">A16</font></i>: 
( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> ) ) &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">x</font> ) &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> holds <br/>( <font color="Maroon" title="c11">FAB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> ) ) )
 
<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" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:114_3_1"/><i><font color="Green" title="E124">A18</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:114_3_1"/><b>then </b><i><font color="Green" title="E125">A19</font></i>: 
<font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:114"><i><font color="Green" title="E119">A13</font></i></a></i>;<br/><a NAME="E3:114_3_1"/>
<font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E4:114_3_1"/>
<font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_3_1"><i><font color="Green" title="E125">A19</font></i></a>, <a class="ref" href="mesfunc5.html#T40" target="_self" title="MESFUNC5:th.40">Th40</a></i>;<br/><b>thus </b> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E17:114"><i><font color="Green" title="E119">A13</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_3_1"><i><font color="Green" title="E124">A18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:114_3_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:114_3_2"/><b>then </b><i><font color="Green" title="E125">A21</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_3"><i><font color="Green" title="E123">A17</font></i></a></i>;<br/><a NAME="E3:114_3_2"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:114_3_2"/><b>then </b>
<font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:114"><i><font color="Green" title="E119">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E5:114_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_3_2"><i><font color="Green" title="E125">A21</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c14">n1</font> = <font color="Maroon" title="c13">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:114_3_3"/>
<font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/><a NAME="E3:114_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E4:114_3_3"/>
<font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:114"><i><font color="Green" title="E119">A13</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c13">n</font>, <font color="Maroon" title="c14">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:114_3_4"/><i><font color="Green" title="E125">A22</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">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" title="c15">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:114_3_4_1"/><i><font color="Green" title="E126">A23</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:114_3_4_1"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E3:114_3_4_1"/><b>then </b><i><font color="Green" title="E127">A24</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:114_3_4_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_3"><i><font color="Green" title="E124">A20</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_3_4_1"><i><font color="Green" title="E126">A23</font></i></a></i>;<br/><b>hence </b><a NAME="E5:114_3_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:114_3_4_1"><i><font color="Green" title="E127">A24</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_3_4"><i><font color="Green" title="E125">A22</font></i></a>, <a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:114_3_4"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 ;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:114_3_5"/><i><font color="Green" title="E125">A25</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:114_3_5"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E3:114_3_5"/><b>then </b><i><font color="Green" title="E126">A26</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div><i><font color="Green" title="E127">A27</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" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_3"><i><font color="Green" title="E124">A20</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_3_5"><i><font color="Green" title="E125">A25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">n</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>
;<br/></div><b>end;</b></div><a NAME="E5:114_3_5"/><b>then </b>
<font color="Maroon" title="c11">FAB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E6:114_3_5"/>
<font color="Maroon" title="c11">FAB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_3_5"><i><font color="Green" title="E126">A26</font></i></a></i>;<br/><b>thus </b> <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> = 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:114_3_5"><i><font color="Green" title="E127">A27</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_3_5"><i><font color="Green" title="E126">A26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_3_5"><i><font color="Green" title="E125">A25</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E22:114"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">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" title="c13">n</font>, <font color="Maroon" title="c14">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_4"/><i><font color="Green" title="E123">A28</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c15">n</font> = <font color="Maroon" title="c13">n</font>, <font color="Maroon" title="c16">m</font> = <font color="Maroon" title="c14">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:114_4"/><i><font color="Green" title="E124">A29</font></i>: 
( <font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E21:114"><i><font color="Green" title="E122">A16</font></i></a></i>;<br/>
<a NAME="E4:114_4"/><i><font color="Green" title="E125">A30</font></i>: 
<font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c16">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:114"><i><font color="Green" title="E121">A15</font></i></a></i>;<br/>
<a NAME="E5:114_4"/><i><font color="Green" title="E126">A31</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:114_4_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_4_1"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:114_4_1"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E21:114"><i><font color="Green" title="E122">A16</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_4"><i><font color="Green" title="E124">A29</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_4"><i><font color="Green" title="E123">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:114_4"/><b>then </b><i><font color="Green" title="E127">A32</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:114_4"><i><font color="Green" title="E124">A29</font></i></a>, <a class="ref" href="mesfunc5.html#T76" target="_self" title="MESFUNC5:th.76">Th76</a></i>;<br/>
<a NAME="E7:114_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:114_4"><i><font color="Green" title="E124">A29</font></i></a>, <a class="txt" href="mesfunc5.html#E5:114_4"><i><font color="Green" title="E126">A31</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self" title="MESFUNC5:th.75">Th75</a></i>;<br/>
<a NAME="E8:114_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c15">n</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E3:114_4"><i><font color="Green" title="E124">A29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:114_4"/>
<font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c14">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:114"><i><font color="Green" title="E121">A15</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_4"><i><font color="Green" title="E125">A30</font></i></a>, <a class="txt" href="mesfunc5.html#E6:114_4"><i><font color="Green" title="E127">A32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:114"/><b>then </b>
<font color="Maroon" title="c12">KAB</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T60" target="_self" title="MESFUNC5:th.60">Th60</a></i>;<br/>
<a NAME="E24:114"/><b>then </b><i><font color="Green" title="E123">A33</font></i>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c12">KAB</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D15" target="_self" title="MESFUNC5:def.15">Def15</a>, <a class="txt" href="mesfunc5.html#E21:114"><i><font color="Green" title="E122">A16</font></i></a>, <a class="txt" href="mesfunc5.html#E20:114"><i><font color="Green" title="E121">A15</font></i></a>, <a class="txt" href="mesfunc5.html#E12:114"><i><font color="Green" title="E116">A10</font></i></a>, <a class="txt" href="mesfunc5.html#E5:114"><i><font color="Green" title="E111">A4</font></i></a>, <a class="txt" href="mesfunc5.html#E13:114"><i><font color="Green" title="E117">A11</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">DA</font> = <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font>;<br/>
<b>set </b><font color="Maroon" title="c14">DB</font> = <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font>;<br/>
<a NAME="E25:114"/>
( <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E26:114"/><b>then </b><i><font color="Green" title="E124">A34</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:114"><i><font color="Green" title="E110">A3</font></i></a>, <a class="ref" href="mesfunc1.html#T34" title="MESFUNC1:th.34">MESFUNC1:34</a></i>;<br/>
<a NAME="E27:114"/><i><font color="Green" title="E125">A35</font></i>: 
( <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp; <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green" title="E109">A2</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E28:114"/><b>then </b>
( <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E29:114"/><b>then </b><i><font color="Green" title="E126">A36</font></i>: 
( <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E30:114"/><b>then </b><i><font color="Green" title="E127">A37</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green" title="E125">A35</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E31:114"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:114_5"/><i><font color="Green" title="E128">A38</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:114"><i><font color="Green" title="E127">A37</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E3:114_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_5"><i><font color="Green" title="E128">A38</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:114"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:114"><i><font color="Green" title="E127">A37</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E33:114"/><b>then </b><i><font color="Green" title="E128">A39</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green" title="E126">A36</font></i></a>, <a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green" title="E124">A34</font></i></a>, <a class="ref" href="mesfunc5.html#T48" target="_self" title="MESFUNC5:th.48">Th48</a></i>;<br/>
<a NAME="E34:114"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:114_6"/><i><font color="Green" title="E129">A40</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:114"><i><font color="Green" title="E127">A37</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E3:114_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_6"><i><font color="Green" title="E129">A40</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:114"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:114"><i><font color="Green" title="E127">A37</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E36:114"/><b>then </b><i><font color="Green" title="E129">A41</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c7">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green" title="E126">A36</font></i></a>, <a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green" title="E124">A34</font></i></a>, <a class="ref" href="mesfunc5.html#T48" target="_self" title="MESFUNC5:th.48">Th48</a></i>;<br/>
<a NAME="E37:114"/><i><font color="Green" title="E130">A42</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:114"><i><font color="Green" title="E108">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font>;<br/>
<b>consider </b><font color="Maroon" title="c15">FA</font> being    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E39:114"/><i><font color="Green" title="E131">A43</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1" title="SEQFUNC:sch.1">SEQFUNC:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font>;<br/>
<b>consider </b><font color="Maroon" title="c16">FB</font> being    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E41:114"/><i><font color="Green" title="E132">A44</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1" title="SEQFUNC:sch.1">SEQFUNC:sch 1</a>();<br/></i>
<a NAME="E42:114"/><i><font color="Green" title="E133">A45</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 
<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" title="c17">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c18">n1</font> = <font color="Maroon" title="c17">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:114_7"/><i><font color="Green" title="E134">A46</font></i>: 
( <font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/>
<a NAME="E3:114_7"/><i><font color="Green" title="E135">A47</font></i>: 
( <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green" title="E131">A43</font></i></a>, <a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green" title="E132">A44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:114_7"/>
( <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T40" target="_self" title="MESFUNC5:th.40">Th40</a>, <a class="txt" href="mesfunc5.html#E2:114_7"><i><font color="Green" title="E134">A46</font></i></a></i>;<br/>

<a NAME="E5:114_7"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:114_7"><i><font color="Green" title="E135">A47</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E6:114_7"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green" title="E125">A35</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green" title="E109">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:114"/><i><font color="Green" title="E134">A48</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">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" title="c17">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c18">n</font> = <font color="Maroon" title="c17">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:114_8"/>
<font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/>
<a NAME="E3:114_8"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E4:114_8"/>
( <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green" title="E131">A43</font></i></a>, <a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green" title="E132">A44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:114"/><i><font color="Green" title="E135">A49</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">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" title="c17">n</font>, <font color="Maroon" title="c18">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_9"/><i><font color="Green" title="E136">A50</font></i>: 
<font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">m</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E2:114_9"/><i><font color="Green" title="E137">A51</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
 ;<br/>

<a NAME="E3:114_9"/><b>then </b>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:114_9"/><b>then </b><i><font color="Green" title="E138">A52</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c17">n</font>, <font color="Maroon" title="c21">m</font> = <font color="Maroon" title="c18">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:114_9"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green" title="E131">A43</font></i></a></i>;<br/>
<a NAME="E7:114_9"/><b>then </b><i><font color="Green" title="E139">A53</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green" title="E133">A45</font></i></a></i>;<br/>
<a NAME="E8:114_9"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green" title="E131">A43</font></i></a></i>;<br/>
<a NAME="E9:114_9"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_9"><i><font color="Green" title="E137">A51</font></i></a>, <a class="txt" href="mesfunc5.html#E7:114_9"><i><font color="Green" title="E139">A53</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E10:114_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_9"><i><font color="Green" title="E136">A50</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_9"><i><font color="Green" title="E138">A52</font></i></a>, <a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:114"/><i><font color="Green" title="E136">A54</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">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" title="c17">n</font>, <font color="Maroon" title="c18">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_10"/><i><font color="Green" title="E137">A55</font></i>: 
<font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">m</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E2:114_10"/><i><font color="Green" title="E138">A56</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span>
 ;<br/>

<a NAME="E3:114_10"/><b>then </b>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:114_10"/><b>then </b><i><font color="Green" title="E139">A57</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c17">n</font>, <font color="Maroon" title="c21">m</font> = <font color="Maroon" title="c18">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:114_10"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green" title="E132">A44</font></i></a></i>;<br/>
<a NAME="E7:114_10"/><b>then </b><i><font color="Green" title="E140">A58</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green" title="E133">A45</font></i></a></i>;<br/>
<a NAME="E8:114_10"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green" title="E132">A44</font></i></a></i>;<br/>
<a NAME="E9:114_10"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c21">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_10"><i><font color="Green" title="E138">A56</font></i></a>, <a class="txt" href="mesfunc5.html#E7:114_10"><i><font color="Green" title="E140">A58</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E10:114_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_10"><i><font color="Green" title="E137">A55</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_10"><i><font color="Green" title="E139">A57</font></i></a>, <a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:114"/><i><font color="Green" title="E137">A59</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> holds <br/>( <font color="Maroon" title="c15">FA</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:114_11"/><i><font color="Green" title="E138">A60</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
 ;<br/>

<a NAME="E2:114_11"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:114_11"/><b>then </b><i><font color="Green" title="E139">A61</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><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" title="c18">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><i><font color="Green" title="E140">A62</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green" title="E131">A43</font></i></a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green" title="E133">A45</font></i></a></i>
;<br/><a NAME="E2:114_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/><a NAME="E3:114_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green" title="E131">A43</font></i></a></i>;<br/><a NAME="E4:114_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_11"><i><font color="Green" title="E138">A60</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_11_1"><i><font color="Green" title="E140">A62</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E5:114_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:114_11"/><b>then </b><i><font color="Green" title="E140">A63</font></i>: 
<font color="Maroon" title="c15">FA</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<a NAME="E6:114_11"/>
( <font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_11"><i><font color="Green" title="E139">A61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:114_11"/>
( <font color="Maroon" title="c15">FA</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:114_11"><i><font color="Green" title="E140">A63</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_11"><i><font color="Green" title="E138">A60</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E47:114"/><i><font color="Green" title="E138">A64</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> holds <br/>( <font color="Maroon" title="c16">FB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:114_12"/><i><font color="Green" title="E139">A65</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span>
 ;<br/>

<a NAME="E2:114_12"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:114_12"/><b>then </b><i><font color="Green" title="E140">A66</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><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" title="c18">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><i><font color="Green" title="E141">A67</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green" title="E132">A44</font></i></a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green" title="E133">A45</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green" title="E132">A44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_12"><i><font color="Green" title="E139">A65</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_12_1"><i><font color="Green" title="E141">A67</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self" title="MESFUNC5:def.13">Def13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E5:114_12"/><b>then </b><i><font color="Green" title="E141">A68</font></i>: 
<font color="Maroon" title="c16">FB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<a NAME="E6:114_12"/>
( <font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_12"><i><font color="Green" title="E140">A66</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:114_12"/>
( <font color="Maroon" title="c16">FB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c17">x</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:114_12"><i><font color="Green" title="E141">A68</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_12"><i><font color="Green" title="E139">A65</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">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" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  =  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c17">KA</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E49:114"/><i><font color="Green" title="E139">A69</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green" title="E140">A70</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" title="c18">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:114_13"/>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:114_13"/>
<font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c18">n</font>)
 <b>by </b><i><a class="txt" href="mesfunc5.html#E49:114"><i><font color="Green" title="E139">A69</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" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  =  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c18">KB</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E52:114"/><i><font color="Green" title="E141">A71</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green" title="E142">A72</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" title="c19">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:114_14"/>
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:114_14"/>
<font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon" title="c19">n</font>)
 <b>by </b><i><a class="txt" href="mesfunc5.html#E52:114"><i><font color="Green" title="E141">A71</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E54:114"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">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" title="c19">n</font>, <font color="Maroon" title="c20">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_15"/><i><font color="Green" title="E143">A73</font></i>: 
<font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font>
 ;<br/>

<a NAME="E2:114_15"/><i><font color="Green" title="E144">A74</font></i>: 
( <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green" title="E133">A45</font></i></a></i>;<br/>
<a NAME="E3:114_15"/><i><font color="Green" title="E145">A75</font></i>: 
( <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E43:114"><i><font color="Green" title="E134">A48</font></i></a></i>;<br/>
<a NAME="E4:114_15"/><i><font color="Green" title="E146">A76</font></i>: 
( <font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> &amp; <font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green" title="E140">A70</font></i></a>, <a class="txt" href="mesfunc5.html#E53:114"><i><font color="Green" title="E142">A72</font></i></a></i>;<br/>
<a NAME="E5:114_15"/><i><font color="Green" title="E147">A77</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:114_15_1"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_15_1"/><b>then </b>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:114_15_1"/><b>then </b>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E44:114"><i><font color="Green" title="E135">A49</font></i></a>, <a class="txt" href="mesfunc5.html#E2:114_15"><i><font color="Green" title="E144">A74</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_15"><i><font color="Green" title="E143">A73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:114_15"/><b>then </b><i><font color="Green" title="E148">A78</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_15"><i><font color="Green" title="E144">A74</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_15"><i><font color="Green" title="E145">A75</font></i></a>, <a class="ref" href="mesfunc5.html#T76" target="_self" title="MESFUNC5:th.76">Th76</a></i>;<br/>
<a NAME="E7:114_15"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_15"><i><font color="Green" title="E144">A74</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_15"><i><font color="Green" title="E145">A75</font></i></a>, <a class="txt" href="mesfunc5.html#E5:114_15"><i><font color="Green" title="E147">A77</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self" title="MESFUNC5:th.75">Th75</a></i>;<br/>
<a NAME="E8:114_15"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E2:114_15"><i><font color="Green" title="E144">A74</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:114_15"/>
<font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green" title="E140">A70</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_15"><i><font color="Green" title="E146">A76</font></i></a>, <a class="txt" href="mesfunc5.html#E6:114_15"><i><font color="Green" title="E148">A78</font></i></a></i>;<br/>

<a NAME="E10:114_15"/><i><font color="Green" title="E149">A79</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:114_15_2"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_15_2"/><b>then </b>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:114_15_2"/><b>then </b>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c21">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E45:114"><i><font color="Green" title="E136">A54</font></i></a>, <a class="txt" href="mesfunc5.html#E2:114_15"><i><font color="Green" title="E144">A74</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_15"><i><font color="Green" title="E143">A73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:114_15"/><b>then </b><i><font color="Green" title="E150">A80</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_15"><i><font color="Green" title="E144">A74</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_15"><i><font color="Green" title="E145">A75</font></i></a>, <a class="ref" href="mesfunc5.html#T76" target="_self" title="MESFUNC5:th.76">Th76</a></i>;<br/>
<a NAME="E12:114_15"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_15"><i><font color="Green" title="E144">A74</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_15"><i><font color="Green" title="E145">A75</font></i></a>, <a class="txt" href="mesfunc5.html#E10:114_15"><i><font color="Green" title="E149">A79</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self" title="MESFUNC5:th.75">Th75</a></i>;<br/>
<a NAME="E13:114_15"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c19">n</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E2:114_15"><i><font color="Green" title="E144">A74</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:114_15"/>
<font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E53:114"><i><font color="Green" title="E142">A72</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_15"><i><font color="Green" title="E146">A76</font></i></a>, <a class="txt" href="mesfunc5.html#E11:114_15"><i><font color="Green" title="E150">A80</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:114"/><b>then </b><i><font color="Green" title="E143">A81</font></i>: 
( ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">m</font> ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">m</font> ) )
 
;<br/>
<a NAME="E56:114"/><b>then </b>
( <font color="Maroon" title="c17">KA</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <font color="Maroon" title="c18">KB</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T60" target="_self" title="MESFUNC5:th.60">Th60</a></i>;<br/>
<a NAME="E57:114"/><b>then </b><i><font color="Green" title="E144">A82</font></i>: 
(  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c17">KA</font> &amp;  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c18">KB</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D15" target="_self" title="MESFUNC5:def.15">Def15</a>, <a class="txt" href="mesfunc5.html#E27:114"><i><font color="Green" title="E125">A35</font></i></a>, <a class="txt" href="mesfunc5.html#E33:114"><i><font color="Green" title="E128">A39</font></i></a>, <a class="txt" href="mesfunc5.html#E36:114"><i><font color="Green" title="E129">A41</font></i></a>, <a class="txt" href="mesfunc5.html#E37:114"><i><font color="Green" title="E130">A42</font></i></a>, <a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="mesfunc5.html#E43:114"><i><font color="Green" title="E134">A48</font></i></a>, <a class="txt" href="mesfunc5.html#E44:114"><i><font color="Green" title="E135">A49</font></i></a>, <a class="txt" href="mesfunc5.html#E46:114"><i><font color="Green" title="E137">A59</font></i></a>, <a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green" title="E140">A70</font></i></a>, <a class="txt" href="mesfunc5.html#E45:114"><i><font color="Green" title="E136">A54</font></i></a>, <a class="txt" href="mesfunc5.html#E47:114"><i><font color="Green" title="E138">A64</font></i></a>, <a class="txt" href="mesfunc5.html#E53:114"><i><font color="Green" title="E142">A72</font></i></a></i>;<br/>
<a NAME="E58:114"/><i><font color="Green" title="E145">A83</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_16"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:114_16"/><i><font color="Green" title="E146">A84</font></i>: 
( <font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green" title="E118">A12</font></i></a></i>;<br/>
<a NAME="E3:114_16"/>
( <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E39:114"><i><font color="Green" title="E131">A43</font></i></a>, <a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green" title="E132">A44</font></i></a></i>;<br/>
<a NAME="E4:114_16"/><b>then </b><i><font color="Green" title="E147">A85</font></i>: 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_16"><i><font color="Green" title="E146">A84</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114"><i><font color="Green" title="E108">A1</font></i></a>, <a class="ref" href="mesfunc5.html#T73" target="_self" title="MESFUNC5:th.73">Th73</a></i>;<br/>
<i><font color="Green" title="E148">A86</font></i>: <font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">n</font> = 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">FAB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:114"><i><font color="Green" title="E121">A15</font></i></a></i>
<br/>.= 
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F0</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E17:114"><i><font color="Green" title="E119">A13</font></i></a></i>
;<br/>
<a NAME="E6:114_16"/>
<font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green" title="E140">A70</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:114_16"/>
<font color="Maroon" title="c12">KAB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:114_16"><i><font color="Green" title="E147">A85</font></i></a>, <a class="txt" href="mesfunc5.html#E5:114_16"><i><font color="Green" title="E148">A86</font></i></a>, <a class="txt" href="mesfunc5.html#E53:114"><i><font color="Green" title="E142">A72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E59:114"/>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">KA</font> implies  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">KB</font> implies  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">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" title="c19">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:114_17_1"/>
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">KA</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c20">n1</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:114_17_1"/><i><font color="Green" title="E146">A87</font></i>: 
( <font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n1</font></span>)</span> &amp; <font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n1</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green" title="E140">A70</font></i></a></i>;<br/><a NAME="E4:114_17_1"/>
<font color="Maroon" title="c15">FA</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n1</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E43:114"><i><font color="Green" title="E134">A48</font></i></a></i>;<br/><b>hence </b><a NAME="E5:114_17_1"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">KA</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:114_17_1"><i><font color="Green" title="E146">A87</font></i></a>, <a class="ref" href="mesfunc5.html#T74" target="_self" title="MESFUNC5:th.74">Th74</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:114_17"/>
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">KB</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">n1</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E4:114_17"/><i><font color="Green" title="E146">A88</font></i>: 
( <font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c20">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n1</font></span>)</span> &amp; <font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n1</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green" title="E133">A45</font></i></a>, <a class="txt" href="mesfunc5.html#E53:114"><i><font color="Green" title="E142">A72</font></i></a></i>;<br/>
<a NAME="E5:114_17"/>
<font color="Maroon" title="c16">FB</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c20">n1</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E43:114"><i><font color="Green" title="E134">A48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:114_17"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">KB</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c19">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:114_17"><i><font color="Green" title="E146">A88</font></i></a>, <a class="ref" href="mesfunc5.html#T74" target="_self" title="MESFUNC5:th.74">Th74</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E60:114"/><b>then </b>
( <font color="Maroon" title="c17">KA</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c18">KB</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self" title="MESFUNC5:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E61:114"/>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E24:114"><i><font color="Green" title="E123">A33</font></i></a>, <a class="txt" href="mesfunc5.html#E57:114"><i><font color="Green" title="E144">A82</font></i></a>, <a class="ref" href="mesfunc5.html#T67" target="_self" title="MESFUNC5:th.67">Th67</a>, <a class="txt" href="mesfunc5.html#E55:114"><i><font color="Green" title="E143">A81</font></i></a>, <a class="txt" href="mesfunc5.html#E58:114"><i><font color="Green" title="E145">A83</font></i></a></i>;<br/>


</div>
