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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/>









<b>assume </b><a NAME="E1:119"/><i><font color="Green">E35</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp;  ex <font color="Olive">A</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">A</font> ) &amp; <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 ;<br/>

<b>consider </b><font color="Maroon">F1</font> being    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">K1</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E3:119"/><i><font color="Green">E37</font></i>: 
( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> ) ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a> ) &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/>( <font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) &amp; <font color="Maroon">K1</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K1</font> )
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:119_1"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">K1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:119_1"/>
( <font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/><b>hence </b><a NAME="E4:119_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T24" target="_self">Th24</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:119"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">K1</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:119"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:119_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:119_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E3:119_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:119_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:119_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:119_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:119_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, , <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:119_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:119_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, </i>;<br/>
<b>hence </b><a NAME="E6:119_2"/>
<font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="txt" href="mesfunc5.html#E12:119"><i><font color="Green">E94</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">A</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:119"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E9:119"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="mesfunc1.html#T41">MESFUNC1:41</a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E10:119"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T1" target="_self">Th1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a>  = <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E12:119"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1">SEQFUNC:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">K</font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E14:119"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:119_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:119_3"/>
<font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E16:119"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_4"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:119_4"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E3:119_4"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/>
<b>hence </b><a NAME="E4:119_4"/>
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E17:119"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_5"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:119_5"/>
<font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E3:119_5"/><b>then </b>
<font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E4:119_5"/>
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:119"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:119_6"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:119_6"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font>, <font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:119_6"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E5:119_6"/><b>then </b><i><font color="Green">E107</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:119_6"/>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E7:119_6"/><b>then </b><i><font color="Green">E108</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E8:119_6"/>
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, , , <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E9:119_6"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="extreal1.html#T42">EXTREAL1:42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:119"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> holds <br/>( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:119_7"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<div><i><font color="Green">E122</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><i><font color="Green">E123</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:119_7_1"><i><font color="Green">E123</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_7_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:119_7_2"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:119_7_2"/>
<font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E4:119_7_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 <b>by </b><i/>;<br/><a NAME="E5:119_7_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E6:119_7_2"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:119_7"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E126</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_7_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:119_7_3"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:119_7_3"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E3:119_7_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E4:119_7_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:119_7_3"><i><font color="Green">E127</font></i></a>, , <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, , <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, </i>;<br/></div><b>end;</b></div>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, , <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E7:119_7"/>
( <font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, , , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:119"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_8"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:119_8"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E3:119_8"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>;<br/>
<b>thus </b><font color="Maroon">K</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> = 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="mesfunc5.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">K1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E21:119"/><b>then </b>
( <font color="Maroon">K</font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">K1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T19" target="_self">Th19</a>, , <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>
<b>hence </b><a NAME="E22:119"/>
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, , <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, , , , , </i>;<br/>


</div>
